Dynamics of Complex Unicritical Polynomials

Dynamics of complex unicritical polynomials

A Dissertation Presented

Davoud Cheraghi

The Graduate School

in Partial Fulﬁllment of the

Requirements

for the Degree of

Doctor of Philosophy

Mathematics

Stony Brook University

August 2009 Stony Brook University

The Graduate School

Davoud Cheraghi

We, the dissertation committee for the above candidate for the Doctor of

Philosophy degree, hereby recommend acceptance of this dissertation.

Mikhail M. Lyubich Professor of Mathematics, Stony Brook University Dissertation Advisor

John Milnor Distinguished Professor of Mathematics, Stony Brook University Chairperson of Defense

Dennis Sullivan Distinguished Professor of Mathematics, Stony Brook University Inside Member

Saeed Zakeri Assitant Professor of Mathematics, City University of New York Outside Member

This dissertation is accepted by the Graduate School.

Lawrence Martin Dean of the Graduate School

ii Abstract of the Dissertation

Dynamics of complex unicritical polynomials

Davoud Cheraghi

Doctor of Philosophy

Mathematics

Stony Brook University

2009

In this dissertation we study two relatively independent problems in one dimensional complex dynamics. One on the parameter space of unicritical polynomials and the other one on measurable dynamics of certain quadratic polynomials with positive area Julia sets. It has been conjectured that combinatorially equivalent non-hyperbolic unicritical polynomials are conformally conjugate. The conjecture has been al- ready established for finitely renormalizable unicritical polynomials. In the first part of this work we prove that a type of compactness, called a priori bounds, on the renormalization levels of a unicritical polynomial, under a certain combinatorial condition, implies this conjecture. It has been recently shown that there are quadratic polynomials with positive area Julia set. The second part of this work investigates typical trajectories of quadratic polynomials with a non-linearizable fixed point of high return

iii type. This class contains the examples with positive area Julia set mentioned above. In particular, we prove that Lebesgue almost every point in the Julia set of these maps accumulates on the non-linearizable ﬁxed point. We show that the post-critical sets of these maps, which are the measure theoretic at- tractors, have measure zero and are connected subsets of the plane. This has some interesting corollaries such as, almost every point in the Julia set of such maps is non-recurrent, or, there is no ﬁnite absolutely continuous invariant measure supported on the Julia set of these maps.

iv Contents

ListofFigures...... vii Acknowledgements ...... viii

1 Introduction 1

2 Preliminaries 11 2.1 JuliaandFatousets ...... 11 2.2 Dynamics of polynomials ...... 14 2.3 Unicritical family and the connectedness locus ...... 15 2.4 Quasi-conformalmappings ...... 20 2.5 Polynomial-like maps ...... 22 2.6 Holomorphicmotions...... 24

3 Combinatorial Rigidity in the Unicritical Family 25 3.1 Modiﬁedprincipalnest ...... 25 3.1.1 Yoccozpuzzlepieces ...... 25 3.1.2 The complex bounds in the favorite nest and renormalization...... 26 3.1.3 Combinatoricsofamap ...... 32 3.2 Proofoftherigiditytheorem...... 35 3.2.1 Reductions ...... 35

v 3.2.2 Thurstonequivalence ...... 37

3.2.3 The domains Ωn,j and the maps hn,j ...... 41

i 3.2.4 The gluing maps gn(k) ...... 58 3.2.5 Isotopy...... 62 3.2.6 Promotiontohybridconjugacy ...... 67 3.2.7 Dynamical description of the combinatorics ...... 69

4 Typical Trajectories of complex quadratic polynomials 72 4.1 Accumulationontheﬁxedpoint...... 72 4.1.1 Post-criticalsetasanattractor ...... 72 4.1.2 The Inou-Shishikura class and the near-parabolic renormalization...... 75 4.1.3 Sectorsaroundthepost-criticalset ...... 80 4.1.4 Sizeofthesectors...... 89 4.1.5 PerturbedFatoucoordinate ...... 93 4.1.6 Proof of main technical lemmas ...... 106 4.2 Measureandtopologyoftheattractor ...... 111 4.2.1 Balls in the complement ...... 111 4.2.2 Going down the renormalization tower ...... 116 4.2.3 Going up the renormalization tower ...... 121 4.2.4 Proofofmainresults ...... 132 Bibliography ...... 135

vi List of Figures

2.1 TheMandelbrotset...... 17 2.2 The connectedness locus and a primary limb in it ...... 18 M3 2.3 A primitive renormalizable Julia set and the corresponding little multibrotcopy ...... 19

3.1 The multiply connected domains and the buﬀers ...... 41 3.2 Primitivecase...... 43 3.3 An inﬁnitely renormalizable Julia set ...... 50 3.4 A twice satellite renormalizable Julia set ...... 53

4.1 A schematic presentation of the Polynomial P ...... 76 4.2 An example of a perturbed Fatou coordinate and its domain. . . 78 4.3 Theﬁrstgenerationofsectors ...... 82

4.4 The domain ΣC2 ...... 100

vii Acknowledgements

First of all, I would like to thank my adviser Mikhail Lyubich for his great patience and support during my graduate studies. He introduced many interesting subjects in holomorphic dynamics to me and was particularly helpful with improving my mathematics writing. Especially, his clear lectures have always been a source of enthusiasm for me and for many others in the dynamics community at the Stony Brook university. I would like to thank the people at the Mathematics Department and IMS at Stony Brook for their generous support. I have enjoyed talking to and learning from many friends there. In particular, I want to thank M. Ander- son, C. Bishop, C. Cabrera, H. Hakobian, M. Martens, J. Milnor, S. Sullivan, S. Sutherland, S. Zakeri. R. Roeder was particularly helpful with improving my mathematics and English writing. I had privilege of spending a year of my studies at the Fields Institute in Toronto. I would like to thank them for their support. Specially, many lengthy conversations with C. Chamanara, J. Kahn, R. Perez, D. Schleicher, M. Shishikura, M. Yampolsky at the Institute created excellent memories during that year. Many thanks go to my great family to whom I owe every thing I have accomplished. Especially, to my parents for their trust in me and the freedom they gave me during all my life, to my brother and sisters for their constant encouragement and the conﬁdence they gave me. Chapter 1

Introduction

In this work we study two different problems in one dimensional holomorphic dynamics. The first one is on combinatorial rigidity of a class of unicritical maps. The second one describes typical (with respect to the Lebesgue measure) trajectories of certain quadratic polynomials with positive area Julia set. Below we describe the content of these two studies. Part 1. Dynamics of a rational map on the Riemann sphere breaks into two pieces; dynamics on the Fatou set and dynamics on the Julia set. Fatou set of a map is “nicely behaved” part of the dynamics, and it is completely understood through works of Fatou [Fat19], [Fat20a], [Fat20b], Julia [Jul18] and Sullivan [Su85]. The Julia set of a rational map is the “chaotic” part of the dynamics. Substantial work by many people has been devoted to understanding these two sets for different families of rational maps. Basic properties of these sets has been collected in several books and surveys: See for example Beardon [Be91], Milnor [M06], Blanchard [Bl84], and Lyubich [Ly86]. Hyperbolic maps form a class of thoroughly understood and well behaved maps. These are rational maps for which the orbits of all critical points tend to attracting periodic points. For every map in this class, there exists a finite

1 subset of the Riemann sphere such that a full measure set of points on the sphere is attracted to this set under dynamics of the map. One of the main conjectures in holomorphic dynamics, which goes back to Fatou, states that the hyperbolic maps are dense in the space of rational maps of degree d (also in the space of polynomials of degree d). By an approach developed through the work of R. Mãne, P. Sad and D. Sullivan [MSS83] for rational maps to tackle this problem, “studying families of rational maps has been reduced to studying of dynamics of individual maps on their Julia set”. Every polynomial with one critical point, which is referred as unicritical polynomial through this thesis, can be written (up to conformal conjugacy) in the form z zd + c, for some complex number c. If the critical point → 0 tends to infinity under iteration of such a map (attracted to the super- attracting fixed point at infinity), the map is hyperbolic. The Julia set is a Cantor set in this case. Therefore, attention reduces to parameters for which orbit of 0 stays bounded under iteration. The set of such parameters is called the connectedness locus or the Mulribrot set, (the well-known Mandelbrot set corresponds to d = 2) see Figures 2.1 and 2.2. There is a way of defining graded partitions of the Multibrot set into pieces such that dynamics of maps in each piece have some special combinatorial property. All maps in a given piece of partition of a certain level will be called combinatorially equivalent up to that level. Conjecturally, combinatorially equivalent (up to all levels) non-hyperbolic maps in the uniciritcal family are conformally conjugate. As stated in [DH82] for d = 2, this Rigidity Conjecture is equivalent to the local connectivity of the Mandelbrot set (MLC for short) and naturally extends to degree d unicritical polynomials. In [DH82], they prove that MLC implies Density of hyperbolic quadratic polynomials among

2 quadratic polynomials. These discussions have been extended to degree d unicritical polynomials by D. Schleicher in [Sch04]. Roughly speaking, the rigidity conjecture says that combinatorics of a map in this class uniquely determines (ﬁne scale) geometry of the chaotic part of the dynamics of that map. In particular, combinatorics of such map determine the parameter c up to a ﬁnite number of values. A unicritical polynomial f is called Douady-Hubbard renormalizable, or renormalizable for short, if there exist a neighborhood U of the critical point

n and an integer n > 1, such that f ◦ : U V is a degree d proper branched → nk covering with U compactly contained in V , and f ◦ (0) U, for k =0, 1, 2,... . ∈ n 1 By [DH85], such a map f ◦ : U V , denoted by (f), is conjugate to a uni- → R critical polynomial, which we denote it by 1(f), of the same degree as the SR one of f. Moreover, this renormalization, under a certain combinatorial condition which will be clear later, provides a homeomorphism from a subset of the connectedness locus onto all of the connectedness locus. Such homeomorphim is determined up finite number of rotations, however, one can make it unique by marking a particular point on the connectedness locus (which will be made clear later). We will denote the corresponding maximal homeomorphic copy of the connectedness locus (domain of this homeomorphism) within the actual connectedness locus by τ(f). By maximal homeomorphic copy we mean that it is not strictly contained in any other homeomorphic copy except the actual connectedness locus. Given a renormalizable unicritical polynomial f, if (f) is also renormal- R izable we say that f is twice renormalizable. So, a unicritical polynomial is called infinitely renormalizable if the renormalization process defined above can be carried out infinite number of times, that is, the sequence (f), ( (f)), R R R

3 ( ( (f))),... is well defined. Combinatorics of an infinitely renormalizable R R R unicritical polynomial is defined as the sequence of maximal homeomorphic copies τ(f), τ( 1(f)), τ(( )2(f)),... of the connectedness locus. SR SR In 1990’s, J. C. Yoccoz [H93] proved MLC conjecture at all non-hyperbolic parameter values which are at most finitely renormalizable. He also proved local connectivity of the Julia set for such maps with all periodic points repelling. Degree 2 assumption was essential in his argument. Local connectivity of Julia sets for unicritical degree d polynomials which are at most finitely renormalizable, and with all their periodic points repelling has been established by J. Kahn and M. Lyubich [KL05] in 2005. Their proof is based on “controlling” geometry of a Modified principal nest. The same controlling technique is used to settle the rigidity problem for these parameters in [AKLS05]. The rigidity conjecture for quadratic maps z2 +c with c real is proved inde- pendently by M. Lyubich [Ly97] and J. Graczyk, G. Sw¸aetek´ [GS98]. Recently, the conjecture has been proved for polynomials with all their critical points real and non-degenerate by O. Kozlovski, W. Shen and S. Van Strien in [KSvS07]. An infinitely renormalizable unicritical polynomial satisfies a priori bounds condition, a notion introduced by D. Sullivan [S92], if infimum of moduli of

annuli V U is non-zero, where ( n(f))tn : U V is a renormalization of n \ n R n → n n(f). Here we prove that this property, under certain combinatorial assump- R tion, implies the rigidity conjecture. We say that an infinitely renormalizable map satisfies the secondary limbs condition, denoted by for short, if the SL maximal connectedness loci copies τ(f), τ( 1(f)),... in the definition of the R combinatorics of f belong to finite number of secondary limbs of the connectedness locus.

Theorem 1.1 (Rigidity). If fc is an inﬁnitely renormalizable degree d uni-

4 critical polynomial, with a priori bounds, and satisﬁes condition, then it is SL combinatorially rigid.

This result was proved in part II of [Ly97] for degree 2 polynomials. The main difference between our proof and the one in [Ly97] is in the construction of the Thurston conjugacy. In [Ly97] such a conjugacy is built along the whole principal nest and uses linear growth of moduli along this nest. However, such a growth is not known for arbitrary degree unicritical polynomials. Certain annuli in the modified principal nest introduced in [KL05] have definite module and this helps us to “pass” over principal nest much easier. This makes the whole construction simpler and more general to include arbitrary degree d. To prove this statement, we first construct a quasi-conformal map of the plane which conjugates the two combinatorially equivalent maps on their post- critical sets. It does not respect the dynamics outside of the post-critical sets, but it is in a “right” homotopy class of homeomorphisms of the complex plane minus the post-critical sets. Verifying such a topological property with the rich possibilities of combinatorics of these maps is the source of difficulty in the proof. Being in such a homotopy class enables one to lift this quasi-conformal map infinite number of times (via the two polynomials) and obtain an actual quasi-conformal conjugacy in the limit. Finally, this quasi-conformal conjugacy can be promoted to a conformal conjugacy by an open-closed argument. Different classes of maps are known to enjoy a priori bounds property; real infinitely renormalizable unicritical polynomials (see [LvS98] and [LY97]), primitive infinitely renormalizable maps of bounded type in [K06], parameters under a Decorations condition in [KL06], and in [KL07] under a Molecule condition. The class of maps satisfying the molecule condition contains the primitive infinitely renormalizable parameters of bounded type and the ones

5 satisfying decoration condition. We will see that the combinatorial class SL contains the combinatorial class of all these parameters. Therefore, combining the above theorem with [KL07] we obtain the following:

Corollary 1.2. Inﬁnitely renormalizable, combinatorially equivalent, quadratic polynomials satisfying the molecule condition are conformally equivalent.

Part 2. Let f : Cˆ Cˆ be a rational map of the Riemann sphere. A → central problem in dynamics of a system is to understand behaviour of orbit of a typical point; z, f(z), f 2(z),...,f n(z),... . Indeed, this is a rather old subject in holomorphic dynamics starting with contributions of Koenigs [Ko84], Schröder [Sch71], Böttcher [Bö04] in the late 19th century on the local dynamics of holomorphic maps. Let f : U C C be a holomorphic map defined ⊆ → on some neighborhood of z0 with f(z0) = z0 and multiplier f ′(z0) = λ. If λ = 0, 1, by [Ko84], it is known that there exists a local conformal change | | 1 of coordinate near z , with ϕ(z ) = 0, and ϕ f ϕ− (z) = z λz near 0 0 ◦ ◦ → 2π p i zero. There is a similar normal form by [Le97] and [Bö04] when λ = e q , and λ = 0, respectively. See [M06] for these results and more. This describes orbits near such fixed points. If the multiplier at a fixed point is λ = e2παi with α R Q, f is called lin- ∈ \ earizable at that fixed point, if there exists a conformal change of coordinate on a neighborhood of the fixed point so that in the new system of coordinate the map becomes the linear map z e2παiz. If there is such a domain of lineariza- → tion, the maximal domain of linearizability is called Siegel disk, otherwise the fixed point is called Cremer fixed point. G. Pfeiffer [Pf17] found first examples of rational maps with a non linearizable fixed point, and H. Cremer [Cre38] proved that for a Baire generic choice of α in (0, 1) every rational map with a fixed point of multiplier e2παi is non linearizable. C. L. Siegel [Sie42] showed 6 that for Lebesgue almost every α in (0, 1) every germ with a fixed point of multiplier e2παi is linearizable. However, these results did not cover all rotations. It was apparent from Cremer and Siegel’s proofs that the answer to this problem depends on the arithmetic nature of α. Let

1 α = [a , a , a ,... ] := 1 2 3 a + 1 1 a2+...

pn denote the continued fraction expansion of α, and = [a1, a2, , an] denote qn its rational approximants. Following Siegel’s ideas, Brjuno [Brj71] proved that given approximants pn of α, under the weaker condition qn

∞ log(q ) n+1 < , q ∞ n=1 n every germ with a ﬁxed point of multiplier e2παi is locally linearizable. We say that a real number α is Brjuno, and write α B, if the above series is ﬁnite. ∈ Yoccoz [Yoc95] showed that the Brjuno condition is sharp for the quadratic maps. In other words, if α / B, P (z)= e2παiz + z2 is not linearizable at 0. ∈ α Now assume that f : C C is a quadratic polynomial. As the dynamics → on the Fatou set is very well understood, we have a complete understanding of typical orbits of a map, if Julia set of that map has zero area. Indeed, this is known in several cases including

if f is hyperbolic, •

if f has a parabolic cycle, see [DH82] or [Ly83b], •

if f is at most ﬁnitely renormalizable with all periodic points repelling, • see [Ly91] or [Sh91],

if f has a Siegel disk with rotation number satisfying log a = (√n), • n O see [PZ04], 7 In particular, siegel disks of bounded type have been studied in greater detail. In [Mc98] and [Ya08] two different types of renormalization technique have been used to describe geometry of the coresponding Julia sets and the dynamics of the map on them. However, X. Buff and A. Chéritat [BC05] in 2005, by completing a program initiated by A. Douady, proved that there are quadratic polynomials with a Cremer fixed point and positive area Julia set. This motivates the problem of describing typical orbits of points in the Julia set of such quadratic polynomials. In the presence of a Cremer fixed point, there is only one Fatou component, the basin of infinity, which is the set of points that tend to the attracting fixed point at infinity under iteration. The Julia set is known to have a complicated topology. For example, it is a non locally connected subset of the plane. Furthermore, by Mañé[Ma83], the finite critical point is recurrent and its orbit accumulates on the Cremer fixed point.

Given N > 0, let IrrN denote the set of irrational numbers α = [a1, a2,... ], with a N for i =1, 2,... . Our ﬁrst result in this direction is that i ≥

Theorem 1.3. There exists a constant N such that if α Irr is a non- ∈ N Brjuno number, then orbit of almost every point in the Julia set of Pα(z) = e2παiz + z2 accumulates on the 0 ﬁxed point.

The statement of the above theorem is trivial if the Julia set has zero area. However, the positive area Julia sets constructed by Buﬀ and Ch´eritat fall into class of maps satisfying our conditions in the above theorem. Thus, above theorem applies to these examples. Proof of this theorem uses a Douady-Ghys type renormalization; return map to a certain domain. In a fruitful work by H. Inou and M. Shishikura [IS06] in 2005 this renormalization was developed to study large iterate of maps

8 near fixed points of high return times. They have introduced a compact (in compact-open topology) class of maps with “large enough” domain of definition which is invariant under this renormalization. More precisely, consider P (z)= z(1+z)2 : U C, where U is a certain domain containing the 0 fixed point and → the 1/3 critical point. The Inou-Shishikura class, , is defined as the class of − IS 1 maps P ϕ− : U C, where ϕ : U U is a conformal map with ϕ(0) = 0, ◦ ϕ → → ϕ 2παi ϕ′(0) = e− . They show that there exists a constant N > 0 such that every map in this class with α Irr is infinitely many times renormalizable, and all ∈ N the renormalizations belong to this class. This result has proved to be useful in studying local dynamics of maps with a neutral fixed point. For example, it was used in Buff and Chéritat’s proof of existence of positive area Julia sets to control post-critical set of such maps. The idea of the proof of above theorem is to construct infinitely many “gates” with the 0 fixed point on their boundary such that almost every point in the Julia set has to go through them. We can also control diameter of gates in terms of the Brjuno function and conclude that these diameters shrink to zero once α is a non-Brjuno number. The orbit of the critical point goes through all these gates (without any assumption on the area of the Julia set). Therefore, a corollary of the above theorem is the following:

Corollary 1.4. If α is a non-Brjuno number in IrrN (same N as in the above theorem) then every map in the Inou-Shishikura class with the ﬁxed point of multiplier e2παi at 0 is non-linearizable. In particular, if f is a rational map of the form e2παi P h, with α Irr , where h is an arbitrary rational map ◦ ∈ N of the Riemann sphere satisfying

h(0) = 0, h′(0) = 1, • no critical value of h belongs to U (domain of P ), • 9 Then f is not linearizable at 0.

For a rational map of the Riemann sphere f, the post-critical set (f) PC is deﬁned as closure of orbits of all critical points of f. It is proved by Lyu- bich [Ly83b] that the post-critical set of a rational map is the measure theoretic attractor of points in the Julia set of that map. That is, for every neighborhood of the post-critical set, orbit of almost every point in the Julia set eventually stays in that neighborhood. A central result in our work on this class of quadratic maps is the following:

Theorem 1.5. There exists a constant N such that for every non-Bruno α ∈ IrrN , the post-critical set of Pα is connected and has zero area.

A straight corollary of the above theorem is the following:

Corollary 1.6. Almost every point in the Julia set of Pα with a non-Brjuno α Irr is non-recurrent. In particular, there is no ﬁnite absolutely continu- ∈ N ous invariant measure on the Julia set.

10 Chapter 2

Preliminaries

2.1 Julia and Fatou sets

Let U be an open subset of the Riemann sphere Cˆ and be a family of holo- F morphic maps defined on U. The family is called normal in U, if every F infinite sequence f ∞ in , contains a subsequence which converges uni- { n}n=1 F formly on all compact subsets of U to a holomorphic map defined on U. Given a rational map f : Cˆ Cˆ, we can consider the family of n-fold compositions → n of f by itself, f ∞ , defined on Cˆ. The Fatou set of f, denoted by F (f), { }n=0 is defined as the largest open subset of Cˆ on which the sequence of iterates

n f ∞ is normal. The Julia set J(f) is defined as the complement of the { }n=0 Fatou set. By definition, these two sets are invariant. The post-critical set of f, which plays an important role in understanding the dynamics of f, is defined as

∞ (f) := f n(c), PC ′ n=1 c:f(c)=0 that is, topological closure of forward images f k(c) with k > 0, of critical points of f. 11 A point z Cˆ is called periodic of period p if f p(z)= z and f j(z) = z for ∈ p 1 1, or λ = 1, respectively. It is called super attracting if λ = 0. | | | | | | It is easy to see that every attracting periodic point is contained in the Fatou set and every repelling one is contained in the Julia set. Indeed, by works of Julia and Fatou, the Julia set is equal to topological closure of repelling periodic points.

2πi p A neutral periodic point is called parabolic if its multiplier is e q for a rational number p/q. Parabolic periodic points belong to the Julia set. A ﬁxed point z with multiplier e2παi is called linearizabale, if there exists a change of coordinate φ deﬁned on some neighborhood of z with φ(z) =0, and

1 2παi φ− f φ = z e z ◦ ◦ → on that neighborhood. If there is no such a coordinate, z is called a Cremer fixed point. In the linearizable case, the maximal domain of linearization is called a Siegel disk. Linearizable periodic points and Cremer periodic points are defined similarly. Every linearizable periodic point belongs to Fatou set and every Cremer one belongs to Julia set. It turns out that the linearizability of a neutral fixed point with an irrational rotation depends on the arithmetic nature of the rotation. Every irrational number can be written as a continued fraction of the form 1 1 , a1 + 1 a2+ . .. which produces best rational approximants

pn 1 = 1 , qn a + 1 1 a2+ ... 1 an 12 for every n 1. The following theorem gives a combinatorial condition on the ≥ rotation which implies linearizability at a neutral ﬁxed point.

Theorem 2.1 (Siegel, Brjuno). If ∞ log q n+1 < , q ∞ n=1 n then every holomorphic germ with a ﬁxed point of multiplier e2παi is locally linearizable.

In the other direction we have,

Theorem 2.2 (Yoccoz). If the above sum is divergent, then the quadratic polynomial z e2παiz + z2 is not linearizable at 0. → By a theorem of Sullivan [Su85] every component of Fatou set is eventually periodic. Let U be a periodic component of Fatou set of a rational map f of period p. There are only ﬁve possibilities for f p on U listed below.

1. Supper attracting case: there exists a super attracting periodic point z U, attracting orbit of every point in U under f p. ∈

p 2. Attracting case: there exists a periodic point z U, 0 < (f )′(z) < ∈ | | 1, attracting orbit of every point in U under f p.

p 3. Parabolic case: there exits a periodic point z ∂U with (f )′(z) = 1, ∈ attracting orbit of every point in U under f p.

4. Siegel disk: U is conformally isomorphic to a disk, and f p acts as an irrational rotation on U.

5. Herman Ring: U is conformally isomorphic to an annulus, and f p acts as an irrational rotation on U.

Finally, M. Shishikura [Sh87] proves a conjeture on an upper bound for number of priodic Fatou components. 13 2.2 Dynamics of polynomials

d d 1 Let f : C C be a monic polynomial of degree d, f(z)= z + a z − + + → 1 a , is a super attracting fixed point of f and its basin of attraction is defined d ∞ as D ( )= z C : f n(z) . f ∞ { ∈ → ∞} Its complement is called the filled Julia set; K(f) = C D ( ). The Julia \ f ∞ set J(f) is the common boundary of K(f) and D ( ). It is known that the f ∞ Julia set and the filled Julia set of a polynomial are connected if and only if all critical points stay bounded under iteration. With f as above, there exists a conformal change of coordinate, Böttcher coordinate, B which conjugates f to the dth power map z zd throughout f → some neighborhood of infinity Uf . It is defined as,

Bf : Uf z C : z >rf 1 , →{ ∈ | | ≥ } (2.1) dn n Bf (z) := lim f (z) n →∞ n where d th root is chosen such that Bf is tangent to the identity map at inﬁnity.

d By definition, it satisfies the equivariance relation Bf (f(z)) = (Bf (z)) , and B (z) z as z . f ∼ →∞ In particular, if the filled Julia set is connected, Bf coincides with the Riemann mapping of D ( ) onto the complement of the closed unit disk, f ∞ normalized to be tangent to the identity map at infinity.

θ θ The external ray R = Rf of angle θ is deﬁned as

1 iθ B− re : r

r r The equipotential E = Ef of level r>rf is deﬁned as

1 iθ B− re :0 θ 2π . f { ≤ ≤ } 14 d It follows from equivariance relation that f(Rθ)= Rdθ, and f(Er)= Er . A ray Rθ is called periodic ray of period p if f p(Rθ) = Rθ. A ray is fixed (p = 1) if and only if θ is a rational number of the form 2πj/(d 1). By − definition, a ray Rθ lands at a well defined point z in J(f) if the limiting value of the ray Rθ (as r r ) exists and equals to z. Such a point z J(f) is → f ∈ called the landing point of Rθ. The following theorem characterizes landing points of periodic rays. See [DH82] for further discussions.

Theorem 2.3. Let f be a polynomial of degree d 2 with connected Julia ≥ set. Every periodic ray lands at a well deﬁned periodic point which is either repelling or parabolic. Vice versa, every repelling or parabolic periodic point is the landing point of at least one, and at most ﬁnitely many periodic rays with the same ray period.

In particular, this theorem implies that the external rays landing at a pe-

p 1 riodic point are organized in several cycles. Let a = a − be a repelling or { k}k=0 parabolic cycle of f and let R(ak) be union of all external rays landing at ak. The conﬁguration p 1 − R(a)= (R(ak)) k=0 with the rays labeled by their external angles, is called the periodic point portrait of f associated to the cycle a.

2.3 Unicritical family and the connectedness

locus

Any degree d polynomial with only one critical point is aﬃnely conjugate to

d Pc(z) = z + c for some complex number c. A case of especial interest is the

15 2πj/(d 1) following fixed point portrait. The d 1 fixed rays R − land at d 1 fixed − − points called βj, and moreover, these are the only rays that land at βj’s. These fixed points are non-dividing, that is, K(F ) β is connected for every j. If the \ j other fixed point, called α, is also repelling, there are at least 2 rays landing at it. Thus, α-fixed point is dividing and by Theorem 2.3, the rays landing at α-fixed point are permuted under the dynamics. The following statement has been shown in [M95] for quadratic polynomials. The same ideas apply to prove it for degree d unicritical polynomials.

Proposition 2.4. If at least 2 rays land at the α ﬁxed point of Pc, we have:

1 The component of C P − (R(α)) containing the critical value is a sector • \ c bounded by two external rays.

1 The component of C P − (R(α)) containing the critical point is a region • \ c bounded by 2d external rays landing in pairs at the points e2πj/dα, for j =0, 1,...,d 1. − The Connectedness locus of degree d is deﬁned as the set of parameters Md c in C for which J(Pc) is connected, or equivalently, the critical point of Pc does not escape to inﬁnity under iteration of P . In particular, is the well- c M2 known Mandelbrot set. See Figures 2.1 and 2.2. A well-known result due to Douady and Hubbard [DH82] shows that the connectedness loci are connected. Their argument is based on constructing an explicit conformal isomorphism

B : C d z C : z > 1 Md \M →{ ∈ | | }

given by B (c)= Bc(c), where Bc is the B¨ottcher coordinate for Pc. Md By means of conformal isomorphism B , the parameter external rays Md

Rθ and equipotentials Er are deﬁned, similarly, as the B -preimages of the Md straight rays going to inﬁnity and round circles around 0. 16 Figure 2.1: The Mandelbrot set

A polynomial Pc (and the corresponding parameter c) is called hyperbolic if P has an attracting periodic point. The set of hyperbolic parameters in , c Md topologically open by deﬁnition, is the union of some components of int . Md These components are called hyperbolic components. The main hyperbolic component is deﬁned as the set of parameter values c

for which Pc has an attracting ﬁxed point. Outside of the closure of this set all ﬁxed points become repelling. Now, consider a hyperbolic component H ⊂ int , and suppose b is the corresponding attracting cycle with period k. On Md c the boundary of H this cycle becomes neutral, and there are d 1 parameters − c ∂H where P has a parabolic cycle with multiplier equal to one. One of i ∈ ci these parameters ci, denoted by croot, which divides the connectedness locus into two pieces, is called root of H (See [DH82] for quadratic polynomials and [Sch04] for arbitrary degree unicritical polynomials). Indeed, any hyperbolic component has one root and d 2 co-roots. The root is the landing point of two − parameter rays, while every co-root is the landing point of a single parameter ray, See Figure 2.3.

17 Figure 2.2: Figure on the left shows the connectedness locus . The ﬁgure M3 on the right is an enlargement of a primary limb in . The dark regions M3 show some of the secondary limbs

Let c belong to a hyperbolic component H , diﬀerent from main component ¯ of the connectedness locus, with attracting cycle bc. The basin of attraction of ¯b , denoted by A , is deﬁned as the set of points z C with P n(z) converges c c ∈ c to the cycle bc. The boundary of the component of Ac containing c is a jordan

k curve which we denote it by Dc. The map Pc on Dc is topologically conjugate to θ dθ on the unit circle. Therefore, there are d 1 ﬁxed points of P k on → − c this jordan curve which are repelling periodic points (of Pc) of period dividing

period of ¯bc (its period can be strictly less than period of bc). Among all rays

landing at these repelling periodic points, let θ1 and θ2 be the angles of the

external rays bounding the sector containing the critical value of Pc (See Figure 2.3). The following theorem makes a connection between external rays Rθ1 ,

θ2 R and the parameter external rays Rθ1 , Rθ2 on the parameter plane. See [DH82] or [Sch04] for proofs.

18 Rθ3 -ray landing at a co-root Rθ1 Rθ2 Rθ1 -ray landing at a root

Rθ2 -ray landing at a root

Figure 2.3: Figure on the left shows a primitive renormalizable Julia set and the external rays Rθ1 and Rθ2 landing at the corresponding repelling periodic point. The ﬁgure on the right is the corresponding primitive little multibrot copy. It also shows the parameter external rays Rθ1 and Rθ2 landing at the root point.

Theorem 2.5. The parameter external rays Rθ1 and Rθ2 land at the root point of H , and these are the only rays that land at this point.

Closure of the two parameter external rays Rθ1 and Rθ2 cut the plane into two components. The one containing the component H , with the root point

attached to it, is called the wake WH . So, a wake is an open set with a root

point attached to its boundary. For a wake WH and an equipotential of level

η, E , the truncated wake WH (η) is the bounded component of WH E . η \ η Part of the connectedness locus contained in the wake WH with the root point

attached to it is called the limb LH of the connectedness locus originated

at H . In other words, the limb LH is part of contained in WH . By Md 19 deﬁnition, every limb is a closed set. The wakes attached to the main hyperbolic component of are called Md primary wakes and a limb associated to such a primary wake will be called primary limb. If H is a hyperbolic component attached to the main hyperbolic component, all the wakes attached to such a component H (except WH itself) are called secondary wakes. Similarly, a limb associated to a secondary wake will be called secondary limb. A truncated limb is obtained from a limb by removing a neighborhood of its root. Some secondary limbs are shown in Figure 2.2.

Given a parameter c in H , we have the attracting cycle bc as above, and the

θ1 associated repelling cycle ac which is the landing point of the external rays R and Rθ2 . The following result gives the dynamical meaning of the parameter values in the wake WH which is bounded by parameter external rays Rθ1 and

Rθ2 (See [Sch04] for the proof).

Theorem 2.6. For parameters c in WH root point , the repelling cycle a \{ } c stays repelling and moreover, the isotopic type of the ray portrait R(ac) is ﬁxed

throughout WH .

2.4 Quasi-conformal mappings

There are several equivalent definitions of quasi-conformal mappings conve- nient in different situations. A good source on the subject is Ahlfor’s book [Ah66]. An analytic definition is more appropriate for us. Let Ω be an open subset of the complex plane. A homeomorphism φ : Ω C is called absolutely → continuous on lines if for every rectangle R contained in Ω with sides parallel to x and y axis, φ is absolutely continuous on almost every horizontal and almost every vertical line in R. Such function has partial derivatives almost 20 every where in Ω. Let φ denote 1 (φ iφ ), and φ denote 1 (φ + iφ ), with z 2 x − y z¯ 2 x y z = x + iy. An orientation preserving homeomorphism φ : Ω C is called K-quasi- → conformal, K-q.c. for short, if

φ is absolutely continuous on lines, •

K 1 φ k φ almost every where in Ω, where k = − . • | z¯| ≤ | z| K+1

The measurable function φ = φz¯/φz is called complex dilatation of φ and the smallest value of K satisfying above deﬁnition is called dilatation of φ. Some properties of q.c. mappings which will be used in our work is listed in the following theorem. One may refer to [Ah66] for their proofs.

Theorem 2.7. Quasi-conformal mappings satisfy the following properties:

Composition of a K -q.c. and a K -q.c. map is a K K -q.c. map. • 1 2 1 2

The space of K-q.c. mappings from Cˆ to Cˆ ﬁxing three points 0, 1, and • is a compact class (under uniform convergence on compact sets). ∞

A 1-q.c. map is conformal. •

We will be interested in solving Beltrami equation φ = for a given measurable function : Cˆ Cˆ. Solution to this problem is referred to as → measurable Riemann mapping theorem.

Theorem 2.8. For any measurable : Cˆ Cˆ with < 1, there exists a → ∞ unique normalized q.c. mapping φ with complex dilatation that leaves 0, 1, fixed. ∞ Any topological annulus is conformally isomorphic to one of C 0 , B(0, 1) \{ } \ 0 , or B(0,r) B(0, 1) for some r > 1. It’s modulus, denoted as mod A, is { } \ 21 defined as in the first two cases and 1 log r in the last case. It turns out ∞ 2π that this quantity is quasi-invariant under q.c. mappings. That is, given a K-q.c. mapping φ : A C, we have 1 mod A mod φ(A) K mod A. → K ≤ ≤ The following theorem makes the connection between q.c. mappings on Ω and maps on boundary of Ω. For convenience we will assume that Ω is upper half plane.

Theorem 2.9 (boundary correspondence). Let φ be a K-q.c. mapping of upper half plane onto itself which maps to itself. Then φ induces a homeomorphism ∞ h : R R which satisﬁes →

1 h(x + t) h(x) M(K)− − M(K), ≤ h(x) h(x t) ≤ − − for some constant M(K) depending only on K. Vice versa, every homeomorphism h : R R with h( ) = that satisﬁes above inequality for some M, → ∞ ∞ extends to a q.c. mapping of the upper half plane with dilatation depending only on M.

2.5 Polynomial-like maps

A holomorphic proper branched covering of degree d, f : U ′ U, where U and → U ′ are simply connected domains with U ′ compactly contained in U, is called a polynomial-like map. Reader can consult [DH85] for the following material on polynomial-like maps. Every polynomial can be viewed as a polynomial-like map once restricted to an appropriate neighborhood of its ﬁlled Julia set. In what follows we will only consider polynomial-like maps with one branched point of degree d (which is assumed to be at zero after normalization) and refer to them as unicritical polynomial-like maps.

22 The ﬁlled Julia set K(f) of a polynomial-like map f : U ′ U is naturally → deﬁned as

n K(f)= z C : f (z) U ′, n =0, 1, 2,... . { ∈ ∈ } The Julia set J(f) is deﬁned as the boundary of K(f). They are connected if and only if K(f) contains critical point of f. Two polynomial-like maps f and g are said topologically (quasi-conformally,

conformally, aﬃnely) conjugate if there is a choice of domains f : U ′ U and → g : V ′ V and a homeomorphism h : U V (quasi-conformal, conformal, or → → aﬃne isomorphism, respectively) such that h f U ′ = g h U ′. ◦ | ◦ | Two polynomial like maps f and g are hybrid or internally equivalent if there is a q.c. conjugacy h between f and g with ∂h = 0 on K(f). The following theorem due to Douady and Hubbard [DH85] makes the connection between polynomial-like maps and polynomials.

Theorem 2.10 (Straightening). Every polynomial-like map f is hybrid equivalent to (suitable restriction of) a polynomial P of the same degree. Moreover, P is unique up to aﬃne conjugacy when K(f) is connected.

In particular, any unicritical polynomial-like map with connected Julia set corresponds to a unique (up to aﬃne conjugacy) unicritical polynomial z → zd + c with c in the connectedness locus . Note that zd + c and zd + c/λ Md are conjugate via z λz for every d 1th root of unity λ. → − Given a polynomial-like map f : U ′ U, we can consider the fundamental → annulus A = U U ′. It is not canonic because any choice of V ′ ⋐ V such \ that f : V ′ V is a polynomial-like map with the same Julia set will give a → diﬀerent annulus. However we can associate a real number, modulus of f, to any polynomial-like map f as follows:

mod(f) = sup mod(A) 23 where the sup is taken over all possible fundamental annuli A of f. It can be seen from the proof of the straightening theorem that dilatation of the q.c. conjugacy between f and the corresponding polynomial P : z c(f) → zd + c(f) can be controlled by modulus of f.

Proposition 2.11. If mod(f) > 0 then dilatation of the q.c. conjugacy ≥ obtained in the straightening theorem depends only on .

2.6 Holomorphic motions

In this section we consider analytic deformations of sets in Cˆ. Let A be a subset of Cˆ. A holomorphic motion of A parametrized on a complex manifold M is a map Φ : M A Cˆ such that × → For any fixed z A, the map λ Φ(λ, z) is holomorphic in M. • ∈ → For any fixed λ M, the map z Φ(λ, z) is an injection. • ∈ → There exists a point λ M with Φ(λ , z)= z, for every z A. • 0 ∈ 0 ∈ The following remarkable result due to Mané-Sad-Sullivan [MSS83] relates holomorphic dynamics with q.c. mappings.

Theorem 2.12 (λ-Lemma). If Φ: M A Cˆ is a holomorphic motion, then × → Φ has an extension to Φ:˜ M A¯ Cˆ, with × → Φ˜ is a holomorphic motion of A¯. • Each Φ(˜ λ, ): A¯ Cˆ is quasi-conformal. • → An important result about holomorphic motions is Slodkowski’s generalized λ-lemma [Sl91] stating that a holomorphic motion of a set A extends to a holomorphic motion of the whole Riemann sphere.

24 Chapter 3

Combinatorial Rigidity in the Unicritical Family

3.1 Modiﬁed principal nest

3.1.1 Yoccoz puzzle pieces

Recall that for a parameter c outside of the main component of the ∈ Md connectedness locus, P has a unique dividing fixed point α . The q 2 c c ≥ external rays R(αc) landing at this fixed point together with an arbitrary equipotential Er, cut the domain inside Er into q closed topological disks Y 0, j = 0, 1,...,q 1, called puzzle pieces of level zero. That is, Y 0’s are the j − j closures of the bounded components of C Er R(¯α ) α . The main \{ ∪ c ∪{ c}} 0 property of this partition is that Pc(∂Yj ) does not intersect interior of any 0 piece Yi . n Now the puzzle pieces Yi of level or depth n are defined as the closures of n 0 the connected components of f − (int(Yj )). They partition the neighborhood n r of the filled Julia set bounded by equipotential f − (E ) into finite number of

25 closed disks. By deﬁnition all puzzle pieces are bounded by piecewise analytic curves. The label of each puzzle piece is the set of the angles of external rays

bounding that puzzle piece. If the critical point does not land on the αc-ﬁxed

n point, there is a unique puzzle piece Y0 of level n containing the critical point.

The family of all puzzle pieces of Pc of all levels has the following Markov property:

Puzzle pieces are disjoint or nested. In the latter case, the puzzle piece • of higher level is contain in the puzzle piece of lower level.

Image of any puzzle piece of level n 1 is a puzzle piece of level n 1. • − n n 1 Moreover, P : Y Y − is d-to-1 branched covering or univalent, c j → k n depending on whether Yj contains the critical point or not.

On the first level, there are d(q 1)+1 puzzle pieces. One critical piece Y 1, − 0 q 1 ones, denoted by Y 1, attached to the fixed point α , and the (d 1)(q 1) − i c − − 1 1 1 symmetric ones, denoted by Z , attached to P − (α ) α . Moreover f Y i c c \{ c} | 0 1 1 1 1 d-to-1 covers Y1 , f Yi univalently covers Yi+1, for i =1,...,q 2, and f Yq 1 | − | − 1 (d 1)(q 1) 1 q 1 1 r univalently covers, Y − − Z . Thus f (Y ) truncated by f − (E ) is 0 ∪ i=1 i 0 1 1 the union of Y0 and Zi ’s. We will assume after this that P n(0) = α for all n, so that the critical c c puzzle pieces of all levels are well defined. As it will be apparent in a moment, this condition is always the case for renormalizable polynomials.

3.1.2 The complex bounds in the favorite nest and renormalization

For a puzzle piece V 0, let R : Dom R V V denote the first return ∋ V V ⊆ → map to V . It is defined at every points z in V for which there exists a positive 26 integer t with P t(z) int V . Then R (z) is defined as P t(z) where t is the c ∈ V c first positive moment when P t(z) int V . Markov property of puzzle pieces c ∈ implies that any component of Dom RV is contained in V and the restriction of

t this return map (Pc , for some t) to such a component is d-to-1 or 1-to-1 proper map onto V . The component of the Dom RV which contains the critical point is called the central component of RV . If the image of critical point under the first return map belongs to the central component, the return is called central return. The first landing map L to a puzzle piece V 0 is defined at all points V ∋ z C for which there exists an integer t 0 with P t(z) int V . It is the ∈ ≥ c ∈ identity map on the component V and univalently maps each component of

Dom LV onto V . Consider a puzzle piece Q 0. If the critical point returns back to Q under ∋ iteration of P , the central component P Q of R is the pullback of Q by P p c ⊂ Q c along the orbit of the critical point, where p is the first moment when critical orbit enters int Q. This puzzle piece P is called the first child of Q. Recall that P p : P Q is a proper map of degree d. c → The favorite child Q′ of Q is constructed as follows; Let p> 0 be the first moment when Rp (0) int(Q P ) (if it exists) and q > 0 be the first moment Q ∈ \ (if it exists) when Rp+q(0) int P (p + q is the moment of the first return ∈ back to P after the first escape of the critical point from P under iterates of

p+q RQ). Now Q′ is deﬁned as the pullback of Q under RQ containing the critical k p+q point. Markov property of puzzle pieces implies that the map Pc = RQ

(for an appropriate k > 0) from Q′ to Q is proper of degree d. The main property of the favorite child is that the image of the critical point under the

k map P : Q′ Q belongs to the ﬁrst child P . c →

27 Consider a unicritical polynomial Pc with q rays landing at its α ﬁxed point

and form the corresponding Yoccoz puzzle pieces. The map Pc is called satellite renormalizable, or immediately renormalizable if

P lq(0) Y 1, for l =0, 1, 2,.... c ∈ 0

The map P q : Y 1 P q(Y 1) is proper of degree d but its domain is not c 0 → c 0 1 1 compactly contained in its range. By slight “thickening” of Y0 so that Y0 is q 1 q compactly contained in Pc (Y0 ) (see [M00]), Pc can be turned into a unicritical polynomial-like map. Note that the above condition implies that the critical point does not escape its domain of deﬁnition and therefore the corresponding little Julia set is connected.

If Pc is not satellite renormalizable, then there is a ﬁrst moment k such

kq 1 1 1 that Pc (0) belongs to some Zi . Deﬁne Q as the pullback of this Zi under kq 1 Pc . By the above construction we form the ﬁrst child, P , and the favorite child ,Q2, of Q1. Repeating the above process we get a nest of puzzle pieces

Q1 P 1 Q2 P 2 Qn P n (3.1) ⊃ ⊃ ⊃ ⊃⊃ ⊃ ⊃ where P i is the ﬁrst child of Qi, and Qi+1 is the favorite child of Qi. The above process stops if and only if one of the following happens:

The map P is combinatorially non-recurrent, that is, the critical point • c does not return to some critical puzzle piece.

n The critical point does not escape the ﬁrst child P under iterates of R n • Q for some n, or equivalently, returns to all critical puzzle pieces of level bigger than n are central.

In the former case, combinatorial rigidity in the critically non-recurrent case has been taken care of in [M00]. 28 k n n In the latter case, R n = P : P Q (for an appropriate k) is a Q c → unicritical polynomial-like map of degree d with P n compactly contained in Qn. The map P is called primitively renormalizable in this case. Note that the corresponding little Julia set is connected because all the returns of critical point to Qn are central by deﬁnition. A map P : C C is called renormalizable if it is satellite or primitively c → renormalizable. To deal with non-renormalizable and recurrent polynomials, the following a priori bounds has been proved in [AKLS05].

Theorem 3.1. There exists δ > 0 such that for every ε > 0 there exists

n0 = n0(ε) > 0 with the following property. For the nest of puzzle pieces

Q1 P 1 Q2 P 2 Qm P m ⊃ ⊃ ⊃ ⊃⊃ ⊃ ⊃ as above, if mod (Q1 P 1) >ε and n δ. \ 0 \

If a map Pc is combinatorially recurrent, the critical point does not land at α-ﬁxed point, therefore puzzle pieces of all levels are well deﬁned. The

combinatorics of Pc up to level n is an equivalence relation on the set of angles

of puzzle pieces up to level n. Two angles θ1 and θ2 are equivalent if the corresponding rays Rθ1 and Rθ2 land at the same point. One can see that the

n combinatorics of a map up to level n+t determines the puzzle piece Yj of level n containing the critical value f t(0). Two non-renormalizable maps are called combinatorially equivalent if they have the same combinatorics up to arbitrary level n,that is, they have the same set of labels of puzzle pieces and the same equivalence relation on them. Combinatorics of a renormalizable map will be deﬁned in section 3.1.3.

Two unicritical polynomials Pc and Pc˜ with the same combinatorics up to level n are called pseudo-conjugate (up to level n) if there is an orientation 29 preserving homeomorphism H : (C, 0) (C, 0), such that H(Y 0) = Y 0 for → j j j =0, 1, 2,... and H P = P H outside of the critical puzzle piece Y n. A ◦ c c˜ ◦ 0 pseudo-conjugacy H is said to match the Böttcher marking if near infinity it becomes identity in the Böttcher coordinates for Pc and Pc˜. Thus a pseudo- conjugacy is the identity map in the Böttcher coordinate outside of Y n by ∪j j its equivariance property.

m m Let qm and pm be the levels of the puzzle pieces Q and P , that is,

m qm m pm Q = Y0 and P = Y0 . The following statement is the main technical result of [AKLS05] which will be used frequently in our construction.

Theorem 3.2. Assume that a (ﬁnite) nest of puzzle pieces

Q1 P 1 Q2 P 2 Qm P m (3.2) ⊃ ⊃ ⊃ ⊃⊃ ⊃

is obtained for Pc. If Pc˜ is combinatorially equivalent to Pc, then there exists a

m qm K-q.c. pseudo-conjugacy H (up to level qm where Q = Y0 ) between Pc and

Pc˜ which matches the B¨ottcher marking.

Proposition 3.3. Assume that the nest of puzzle pieces in the above theorem is deﬁned using equipotential of level η, then dilatation of the q.c. pseudo- conjugacy, K = K(c, c˜), obtained in that theorem depends only on the distance between c and c˜ in the primary wake truncated by equipotential of level η and equipped with the Poincar´emetric.

Proof. To prove the proposition we need a brief sketch of the proof of the above

theorem. For more details refer to [AKLS05]. Combinatorial equivalence of Pc

and Pc˜ up to level zero implies that the parameters c andc ˜ belong to the same truncated wake W (η) attached to the main component of the Connectedness locus. Inside W (η), the q external rays R(α) and the equipotential E(h) (for

30 every h > η) move holomorphically in C 0. That is, there exists a holomor- \ 1 phic motion Φ of R(α) E(h), given by B− B in the second coordinate, ∪ c˜ ◦ c parametrized on W (η) such that

φ(˜c, R(α) E(h)) = (˜c, R(α) E(h)). ∪ ∪

Outside of equipotential E(h), this holomorphic motion extends to a motion holomorphic in both variables (c, z) which is coming from the B¨ottcher

c˜ c 1 coordinate near . By [Sl91] the map φ (φ )− extends to a K q.c. ∞ ◦ 0 map G : (C, 0) (C, 0), where K only depends on the hyperbolic dis- 0 → 0 tance between c andc ˜ in the truncated wake W (η). This gives a q.c. map G : (C, 0) (C, 0) which conjugates P and P outside of puzzle pieces of 0 → c c˜ level zero.

By adjusting the q.c. map G0 inside equipotential E(h) such that it sends c toc ˜, we get a q.c. map (not necessarily with the same dilatation) G0′ . By ˜ lifting G0′ via Pc and f we get a new q.c. map G1. Repeating this process, for i = 1, 2,...,n = qm, which is adjusting the q.c. map Gi inside union of puzzle pieces of level i + 1 so that it sends c toc ˜ and lifting it, we obtain a q.c. map Gi+1 (not with the same dilatation) conjugating Pc and Pc˜ outside union of puzzle pieces of level i + 1. At the end we will have a q.c. map Gn which

n conjugates Pc and Pc˜ outside of equipotential E(h/d ). The nest of puzzle pieces

Q1 P 1 Q2 P 2 Qm P m ⊃ ⊃ ⊃ ⊃⊃ ⊇ for Pc˜ is deﬁned as the image of the nest of puzzle pieces in (3.2) under the map Gn. Combinatorial equivalence of Pc and Pc˜ implies that this new nest

i+1 has the same properties as the one for Pc. In other words, Q is the favorite child of Qi and P i is the ﬁrst child of Qi. Hence Theorem 3.1, applies to this 31 nest too. By properties of these nests, one constructs a K-q.c. map Hn from the critical puzzle piece Qn to the corresponding one Qn where K only depends on the a priori bounds δ and the hyperbolic distance between c andc ˜ in the η truncated wake W . The pseudo-conjugacy Hn is obtained from univalent lifts

of Hn onto other puzzle pieces.

If Pc is renormalizable, the process of constructing modiﬁed principal nest stops at some level and returns to the critical puzzle pieces after this level are central. One can see that critical puzzle pieces do not shrink to 0.

3.1.3 Combinatorics of a map

If P : z zd +c is renormalizable, there is a homeomorphic copy 1 c of c0 → 0 Md ∋ 0 the connectedness locus within the connectedness locus satisfying the following properties (see[DH85]): For c 1 the root point , P : z zd + c is ∈ Md \{ } c → renormalizable, and there is a holomorphic motion of the dividing fixed point α and the rays landing at it on a neighborhood of 1 the root point , such c Md \{ } that the renormalization of Pc is associated to this fixed point and external rays. In other words, all parameters in this copy have Yoccoz puzzle pieces of all levels with the same labels. This homeomorphism is not unique because of the symmetry in the connectedness locus. However, we define it uniquely by sending the unique root point of the copy to the landing point of the parameter external ray of angle 0.

We show below that among all renormalizations of Pc, there is a unique one denoted by P which corresponds to a maximal copy (not included in any R c other copy except itself) of the connectedness locus inside the connected- Md ness locus.

j Assume that P is equal to P : U U ′, for some positive integer j and R c c → 32 topological disk U compactly contained in U ′. By straightening theorem, P R c is conjugate to a unicritical polynomial Pc′ . That is, there exists a q.c. mapping

j χ : U ′ C, with χ P (z) = P ′ χ(z), for every z U. This polynomial → ◦ c c ◦ ∈ Pc′ is determined up to conformal equivalence in the theorem. However, there are only d 1 conformally equivalent polynomials in each class. We make − this parameter unique by choosing the image of c under the homeomorphism uniquely determined above.

If Pc′ is also renormalizable, Pc is called twice renormalizable. Let positive

k integer k, and topological disks V and V ′ be such that P ′ : V V ′ is a renor- c → malization of Pc′ . Deﬁne V and V ′ as χ-preimage of V and V ′, respectively.

jk k One can see that χ conjugates P : V V ′ with P ′ : V V ′. Therefore, c → c → jk 2 P : V V ′ is also a polynomial-like map. We denote this map by P . c → R c Above process may be continued to associate a canonical finite or infinite sequence P , P , 2P ,..., of polynomial-like maps to P , and accordingly, c R c R c c call Pc at most finitely, or infinitely renormalizable. Equivalently, there is a finite or infinite sequence τ(P ) := 1, 2,... , of maximal connectedness c Md Md locus copies associated to P , where n corresponds to the renormalization c Md n n 1 P of − P . In the infinitely renormalizable situation, τ(P ) is called the R c R c c combinatorics of Pc. Two infinitely renormalizable maps are called combinatorially equivalent if they have the same combinatorics, i.e. correspond to the same sequence of maximal connectedness locus copies.

We say an infinitely renormalizable map Pc satisfies the secondary limbs condition, if all of the connectedness copies in the combinatorics τ(Pc) belong to a finite number of truncated secondary limbs. Let stand for the class SL of infinitely unicritical polynomial-like maps satisfying the secondary limbs

33 condition.

An inﬁnitely renormalizable map Pc has a priori bounds, if there is an ε> 0 with mod ( mP ) ε, for all m 0. R c ≥ ≥

34 3.2 Proof of the rigidity theorem

In this section we begin to prove the rigidity theorem in a slightly more general form as follows.

3.2.1 Reductions

Theorem 3.4 (Rigidity theorem). Let f and f˜ be two inﬁnitely renormalizable unicritical polynomial-like maps satisfying condition with a priori bounds. SL If f and f˜ are combinatorially equivalent, then they are hybrid equivalent.

Remark. If the two maps f and f˜ in the above theorem are polynomials, then hybrid equivalence becomes conformal equivalence. That is because the B¨ottcher coordinate, which conformally conjugates the two maps on the complement of the Julia sets, can be glued to the hybrid conjugacy on the Julia set. See Proposition 6 in [DH85] for a precise proof of this.

The proof breaks into following steps:

combinatorial equivalence

⇓ topological equivalence

⇓ q.c. equivalence

⇓ hybrid equivalence

It has been shown in [J00] that any unbranched inﬁnitely renormalizable map with a priori bounds has locally connected Julia set. A renormalization

35 f n : U V is called unbranched if (f) U = (f n : U V ). Here, → PC ∩ PC → unbranched condition follows from our combinatorial condition and a priori bounds (see [Ly97] Lemma 9.3). Then the first step, topological equivalence of combinatorially equivalent maps, follows from the local connectivity of the Julia sets by the Carathéodory theorem. Indeed by [Do93] there is a topological model for the Julia set of these maps based on their combinatorics. The last step in general follows from McMullen’s Rigidity Theorem [Mc94] (Theorem 10.2). He has shown that an infinitely renormalizable quadratic polynomial-like map with a priori bounds does not have any nontrivial invariant line field on its Julia set. The same proof works for degree d unicritical polynomial-like maps. It follows that any q.c. conjugacy h between f and f˜ satisfies ∂h = 0 almost everywhere on the Julia set. Therefore, h is a hybrid conjugacy between f and f˜. However, if all infinitely renormalizable unicritical maps in a given combinatorial class satisfy a priori bounds condition, it is easier to show that q.c. conjugacy implies hybrid conjugacy for that class rather than showing that there is no nontrivial invariant line field on the Julia set. Since we are finally going to apply our theorem to combinatorial classes for which a priori bounds have been established, we will prove it in Proposi- tion 3.20. So assume that f and f˜ are topologically conjugate. We want to show the following:

Theorem 3.5. Let f and f˜ be inﬁnitely renormalizable unicritical polynomial- like maps satisfying a priori bounds and conditions. If f and f˜ are topo- SL logically conjugate then they are q.c. conjugate.

36 3.2.2 Thurston equivalence

Suppose two unicritical polynomial-like maps f : U U and f˜ : U U 2 → 1 2 → 1 are topologically conjugate. A q.c. map

h :(U , U , (f)) (U , U , (f˜)) 1 2 PC → 1 2 PC is a Thurston conjugacy if it is homotopic to a topological conjugacy

ψ :(U , U , (f)) (U , U , (f˜)) 1 2 PC → 1 2 PC between f and f˜ relative ∂U ∂U (f). Note that a Thurston conjugacy 1 ∪ 2 ∪ PC is not a conjugacy between two maps. It is a conjugacy on the postcritical set and homotopic to a conjugacy on the complement of the postcritical set. We will see in the next lemma that it is in the “right” homotopy class. The following result is due to Thurston and Sullivan [S92] which originates the “Pull-Back Method” in holomorphic dynamics.

Lemma 3.6. Thurston conjugate unicritical polynomial-like maps are q.c. conjugate.

Proof. Assume h :(U , U , (f)) (U , U , (f˜)) is a Thurston conjugacy 1 1 2 PC → 1 2 PC homotopic to a topological conjugacy Ψ : (U , U , (f)) (U , U , (f˜)) 1 2 PC → 1 2 PC relative ∂U ∂U (f). 1 ∪ 2 ∪ PC As f : U 0 U f(0) and f˜ : U 0 U f˜(0) are covering 2 \{ }→ 1 \{ } 2 \{ }→ 1 \{ } maps, h : U f(0) U f˜(0) can be lifted to a homeomorphism 1 1 \ { } → 1 \ { } h : U 0 U 0 . Moreover, since h satisﬁes the equivariance relation 2 2 \{ }→ 2 \{ } 1 h f = f˜ h on the boundary of U , h can be extended onto U U by h . It 1 ◦ ◦ 1 2 2 1 \ 2 1 also extends to the critical point by sending it to the critical point of f˜. Let us denote this new map by h2. For the same reason, every homotopy ht between

37 ˜ Ψ and h1 can be lifted to a homotopy between Ψ and h2. As f and f are holomorphic maps, h2 has the same dilatation as dilatation of h1. This implies

that the new map h2 is also a Thurston conjugacy with the same dilatation as

the one of h1. By deﬁnition, the new map h2 satisﬁes the equivariance relation

1 on the annulus U f − (U ). 2 \ 2 Repeating the same process with h2, we obtain a q.c. map h3 and so on.

Thus, we have a sequence of K-q.c. maps hn from U1 to U1 which satisﬁes

n equivariance relation on the annulus U f − (U ). All these maps can be 2 \ 2 extended onto complex plane (see Theorem 2.9) with a uniform bound on their dilatation. This family of q.c. maps is normalized at points 0, f(0), f 2(0),... by mapping them to the corresponding points 0, f˜(0), f˜2(0),... . Compactness of

this class, Theorem 2.7, implies that there is a subsequence hnj which converges

to a K-q.c. map H on U1.

For every z outside of the Julia set, the sequence hnj (z) stabilizes and, by definition, eventually h f(z) = f h (z). Taking limit of both sides nj ◦ ◦ nj will imply that H f(z)= f H(z) for every such z. As filled Julia set of an ◦ ◦ infinitely renormalizable unicritical map has empty interior, conjugacy relation for an arbitrary z on the Julia set follows from continuity of H.

By the a priori bounds assumption in the theorem, there are topological

disks V ⋐ U containing 0 such that nf = f tn : V U is a unicritical n,0 n,0 R n,0 → n,0 degree d polynomial-like map and mod (U V ) ε. By going several levels n,0 \ n,0 ≥ tn ktn ktn down, i.e. considering f : f − (V ) f − (U ) for some positive integer n → n k, we may assume that mod (U V ) and mod (U V ) are proportional. n \ n n \ n Also by slightly shrinking the domains, if necessary, we may assume that these domains have smooth boundaries. Thus we have

- For every n 1, we have mod (U V ) ε,and mod (U V ) ε, ≥ n,0\ n,0 ≥ n,0 \ n,0 ≥ 38 - There exits a constant M such that for every n 1, ≥ 1 mod (U V ) n \ n M, M ≤ mod (U V ) ≤ n \ n - For every n 1, U and U have smooth boundaries. ≥ n,0 n,0 We use the following notations throughout the rest of this note. f : V U ,K = K(f), 0 → 0 0,0 f = f t1 : V U ,K = K( f), R 1,0 → 1,0 1,0 R 2f = f t2 : V U ,K = K( 2f), R 2,0 → 2,0 2,0 R . .

nf = f tn : V U ,K = K( nf), R n,0 → n,0 n,0 R . .

The domain V , for i = 1, 2,...,t 1, is deﬁned as the pullback of n,i n − i tn i Vn,0 under f − containing the little Julia set Jn,i := f − (Jn,0) and Un,i as the

i tn component of f − (U ) containing V so that f : V U is a polynomial- n,0 n,i n,i → n,i like map. The domain Wn,i is deﬁned as the preimage of Vn,i under the map

f tn : V U . n,i → n,i i Accordingly, Kn,i is defined as the component of f − (Kn,0) inside Vn,i. Note that nf : V U is a polynomial-like map with the filled Julia R n,i → n,i set K which is conjugate to nf : V U by conformal isomorphism n,i R n,0 → n,0 f i : U U . It has been proved in [Ly97] (Lemma 9.2) that there is al- n,i → n,0 ways definite space in between Julia sets in the primitive case for parameters under our assumption. Compare the proof of Lemma 3.13. It has been shown in [Mc96] that definite space between little Julia sets implies that there exist

choice of domains Un,i which are disjoint for different i’s and moreover the annuli U V have definite moduli. So we will assume that on the primitive n,i \ n,i levels, the domains Un,i are disjoint for different i’s. 39 In all of the above notation, the first lower subscripts denote the level of renormalization and the second lower subscripts run over little filled Julia sets, Julia sets and their neighborhoods accordingly. In what follows all corresponding objects for f˜ will be marked with a tilde and any notation introduced for f will be automatically introduced for f˜ too. To build a Thurston conjugacy, we first introduce multiply connected domains Ωn(k),i (and Ωn(k),i) in C for an appropriate subsequence n(k) of the renormalization levels and a sequence of q.c. maps with uniformly bounded distortion h : Ω Ω n(k),i n(k),i → n(k),i for k =0, 1, 2,... and i =0, 1, 2,...,t 1. These domains will satisfy the n(k) − following properties:

Each Ω , for n(k) 0, is a topological disk minus n(k+1) topological • n(k),j ≥ disks Dn(k+1),i.

Each Ω , for n(k) 1, is well inside D which means that the mod- • n(k),i ≥ n(k),i ulus of the annulus obtained from D Ω is uniformly bounded n(k),i \ n(k),i below for all n(k) and i.

Every little postcritical set J (f) is well inside D . • n(k),i ∩ PC n(k),i

i Every D is the pullback of D under f − containing J (f) • n(k),i n(k),0 n(k),i∩PC i and every Ωn(k),i is the component of f − (Ωn(k),0) inside Dn(k),i.

Finally, we construct a Thurston conjugacy by an appropriate gluing of the maps h : Ω Ω together on the complement of all these multiply n(k),i n(k),i → n(k),i connected domains (which is a union of annuli). See Figure 3.1.

40 Dl,0 Ωl,0 Dl+1,0 Ωl+1,0 D˜ l,0 Ω˜ l,0 D˜ l+1,0 Ω˜ l+1,0

b b b b gl b b

hl,0

gl+1 hl+1,0

Figure 3.1: The multiply connected domains and the buﬀers

3.2.3 The domains Ωn,j and the maps hn,j

By applying the straightening theorem to the polynomial-like map

n 1 − f : Vn 1,0 Un 1,0 R − → −

we get a K1(ε)-q.c. map and a unicritical polynomial fcn−1 , such that

0 1 Sn 1 :(Un 1,0,Vn 1,0, 0) (Υn 1, Υn 1, 0), (3.3) − − − → − − n 1 Sn 1 − f = fcn−1 Sn 1. − ◦ R ◦ − See Figure 3.2.

Remark. To make notations easier to follow, we will drop the second subscript whenever it is zero and it does not create any confusion. Also, all objects

on the dynamic planes of fcn−1 and fc˜n−1 (the ones after straightening) will be denoted by bold face of notations used for objects on the dynamic planes of f and f˜.

To deﬁne Ωn 1,j and hn 1,j, because of of the diﬀerence in type of renor- − − malizations, we will consider the following three cases: 41 n 1 A . − f is primitively renormalizable. R

n 1 n B. − f is satellite renormalizable and f is primitively renormalizable. R R

n 1 n C . − f is satellite renormalizable and f is also satellite renormalizable. R R

For a given inﬁnitely renormalizable map fc, the renormalization on each level is of primitive or satellite type. Therefore, we can associate a word

P...PS...SP...

of P and S where a P or a S in the i’s place means that the i’s renormaliza-

tion of fc is of primitive or satellite type, respectively. Corresponding to any

m1 m2 m3 such word, we deﬁne a word of cases A B C ... (for non-negative mj’s) deﬁned as follows. Starting from left, a P is replaced by A , SP by B, and SS by C S. By repeating this process, we obtain a word of cases which is used to decide which case to pick at each step.

Case A :

We need the following lemma to show that there are equipotentials of suﬃ-

ciently high level η(ε) inside Sn 1(Wn 1,0) and S˜n 1(Wn 1,0) in the dynamic − − − −

planes of the maps f − and f − . cn 1 c˜n 1

Lemma 3.7. If P : U ′ U is a unicritical polynomial with connected Julia c → set and mod (U U ′) ε, then U ′ contains equipotentials of level less than \ ≥ η(ε) depending only on ε.

Proof. The map Pc on the complement of K(Pc) is conjugate to P0 on the

complement of the closed unit disk D1 by B¨ottcher coordinate Bc. Since levels of equipotentials are preserved under this map and modulus is conformal invariant, it is enough to prove the statement for P : V ′ V for V ′ compactly 0 → 42 Rn−1f n−1 ˜ hn−1,0 R f

n−2 R f Rn−2f˜

Un−1,0 ˜ Un−1,0

˜ Sn−1 Sn−1

Υ0 n−1,0 Υ1 n−1,0

f f cn−1 c˜n−1 hn−1,0

˜1 Qn−1

1 Qn−1,0

Figure 3.2: Primitive case

1 contained in V and mod (V V ′) ε. As P : P − (V V ′) (V V ′) is a \ ≥ 0 0 \ → \ 1 covering of degree d, modulus of the annulus P − (V V ′) is ε/d which implies 0 \ that modulus of V ′ D ε/d. By Grötzsch problem in [Ah66] (Section A in \ 1 ≥ Chapter III) we conclude that V ′ D contains a round annulus D D . \ 1 η(ε) \ 1 By considering the equipotentials of level η(ε) (obtained in the previous lemma) and the external rays landing at the dividing fixed points αn 1 andα ñ 1 − − of the maps fcn−1 , and fcñ−1 , we can form the favorite nest of puzzle pieces (3.1) introduced in Section 3.1.2. The hyperbolic distance between cn 1 andc ñ 1 − − in the truncated primary wake containing cn 1 andc ñ 1, W (η(ε)), is bounded − − by some M(ε) depending only on ε and the combinatorial class . That SL 43 is because cn 1 andc ñ 1 belong to a finite number of truncated limbs which − − is a compact subset of this wake. Therefore, by Proposition 3.3, dilatation of the pseudo-conjugacy obtained in Theorem 3.2 is uniformly bounded by some constant K2(ε).

χn qχn χn Let Qn,0 = Yn,0 and Pn,0 denote the last critical puzzle pieces obtained in the nest (3.1), and hn 1 = hn 1,0 denote the corresponding K2(ε)-q.c. pseudo- − − i χn i χn conjugacy obtained in Theorem 3.2. Components of fc−n−1 (Qn,0) and fc−n−1 (Pn,0) containing the little Julia sets Jn,i , for i =0, 1, 2,...,tn/tn 1 1, are denoted − − χn χn by Qn,i and Pn,i , respectively. Note that that tn/tn 1 is the period of the ﬁrst − renormalization of fcn−1 .

As the polynomials fcn−1 and fc˜n−1 also satisfy our combinatorial condition and a priori bounds assumption, there is a topological conjugacy, denoted by

1 ψn 1, between them which is obtained from extending Bc˜− −1 Bcn−1 onto the − n ◦ Julia set.

χn χn Now we would like to adjust hn 1 : Qn Qn using dynamics of − →

f tn/tn−1 : P χn Qχn , and f tn/tn−1 : P χn Qχn . cn−1 n → n c˜n−1 n → n

Let A0 denote the closure of the annulus Qχn P χn , and Ak , for k =0, 1, 2,... , n n \ n n ktn/tn−1 0 k denote the component of fc−n−1 (An) around Jn,0 by An. We can lift hn 1 by − tn/tn−1 χn tn/tn−1 χn 0 0 fc −1 : Q C and f : Q C to obtain a K -q.c. map g : A A n n → c˜n−1 n → 2 n → n 0 which is homotopic to ψn 1 relative boundaries of An. That is because by the − χn χn 0 external rays connecting ∂Pn to ∂Qn , the annulus An is partitioned into some topological disks and the two maps coincide on the boundaries of these topological disks.

ktn/tn−1 k 0 ktn/tn−1 k 0 As fc −1 : A A and f : A A are holomorphic un- n n → n c˜n−1 n → n branched coverings, g can be lifted to a K -q.c. map from Ak to Ak , for every 2 n n k 1. All these lifts are the identity map in the B¨ottcher coordinate on the ≥ 44 boundaries of these annuli. Hence, they match together to K2-q.c. conjugate the two maps

f tn/tn−1 : P χn J Qχn J , and f tn/tn−1 : P χn J˜ Qχn J˜ . cn−1 n \ n → n \ n cñ−1 n \ n → n \ n Finally, we would like to extend this map further onto little Julia set Jn. This is a especial case of a more general argument presented below. Given a polynomial f with connected Julia set J, the rotation of angle θ on C J is defined as the rotation of angle θ in the Bötcher coordinate on C J, \ \ 1 iθ that is, B− (e B ). By means of straightening, one can define rotations on c c the complement of the Julia set of a polynomial-like map. It is not canonical as it depends on the choice of straightening map. However, its effect on the landing points of external rays is canonical.

Proposition 3.8. Let f : V V be a polynomial-like map with connected 2 → 1 Julia set J. If φ : V J V J is a homeomorphism which commutes with 1 \ → 1 \ f, then there exists a rotation of angle 2πj/(d 1), ρ , such that ρ φ extends − j j ◦ as the identity map onto J.

Proof. Consider an external ray R landing at the non-dividing ﬁxed point

β0 of f. As this ray is invariant under f and φ commutes with f, we have f φ(R) = φ f(R) = φ(R). Thus, φ(R) is also invariant under f which ◦ ◦ implies that φ(R) lands at a non-dividing ﬁxed point βj of f. Now, choose ρj such that ρ (φ(R)) lands at β . Let ψ denote ρ φ and R′ denote ρ (φ(R)). j 0 j ◦ j For such a rotation ρj, ψ also commutes with f and R′ is also invariant under f. The external ray R cuts the annulus V V into a quadrilateral I . The 1 \ 2 0,1 1 preimage f − (I0,1) produces d quadrilaterals denoted by

I1,1,I1,2,...,I1,d, 45 n ordered clockwise starting with R. Similarly, the f -preimage of I0,1 produces

n d quadrilaterals In,1,In,2,...,In,dn (also ordered clockwise starting with R).

In the same way, the external ray R′ produces quadrilaterals denoted by In,j′ ordered clockwise starting with R′. First we will show that the Euclidean diameter of In,j (and In,j′ ) goes to zero as n tends to inﬁnity. i Denote f − (V1) by Vi+1 and let di+1 denote the hyperbolic distance on the annulus V J. As I V , and the intersection of the nest of topological i+1 \ n,j ⊆ n+1 disks V is equal to J, the quadrilaterals I converge to the boundary of V J n n,j 1 \ as n goes to inﬁnity. In order to show that the Euclidean diameter of these quadrilaterals go to zero, it is enough to show that their hyperbolic diameters in

n 1 (V J, d ) stay bounded. Since f − :(V J, d ) (V J, d ) is an unbranched 1\ 1 n\ n → 1\ 1 n 1 n 1 covering of degree d − , therefore an isometry, and closure of f − (In,j) is a compact subset of V J, we conclude that I has bounded hyperbolic diameter 1\ n,j in (V J, d ). Finally, contraction of the inclusion map from (V J, d ) to n \ n n \ n (V J, d ) implies that I has bounded hyperbolic diameter in (V J, d ). 1 \ 1 n,j 1 \ 1 With the same type of argument, one can show that the hyperbolic distance between I and I′ inside (V J, d ) is also uniformly bounded. n,j n,j 1 \ 1

Since ψ is a conjugacy it sends In,j to In,j′ , and therefore, as w converges to

J, w and ψ(w) belong to In,j and In,j′ with larger and larger index n. Combining with the above argument, we conclude that the Euclidean distance between these two points tends to zero. This implies that ψ can be extended as the identity map on the Julia set.

A particular case of above proposition is that every homeomorphism which commutes with a quadratic polynomial-like map can be extended onto Julia set.

1 χn χn Applying the above lemma to ψn− 1 g with V1 = Qn , V2 = Pn , and an − ◦

46 χn external ray connecting ∂Qn to Jn,0 we conclude that g can be extended as

ψn 1 onto Jn,0. It also follows from proof of the above lemma that g and ψn 1 − − are homotopic relative J ∂Qχn . That is because the quadrilaterals obtained n ∪ n χn in the above lemma cut the puzzle piece Qn into inﬁnite number of topological disks such that g and ψn 1 are equal on their boundaries. − χn Similarly, hn 1 can be adjusted on the other puzzle pieces Qn,i, and more- − χn χn over, hn 1 is homotopic to ψn 1 on Qn,i relative Jn,i ∂Qn,i. We will denote − − ∪ the map obtained from extending hn 1 onto little Julia sets Jn,i with the same − notation hn 1. −

The ﬁnal argument is to prepare hn 1 for the next step of the process. It − is stated in the following lemma.

Lemma 3.9. The map hn 1 can be adjusted (through a homotopy) on a neigh- − borhood of iJn,i to a q.c. map hn′ 1 which maps Vn,i = Sn 1(Vn,i) onto ∪ − − ˜ Vn,i = Sn 1(Vn,i). Moreover, dilatation of hn′ 1 is uniformly bounded by a − − constant K (ε) depending only on ε. 2

Proof. The basic idea is move all of hn 1(Vn,i) simultaneously close enough to − little Julia sets Jn,i, and then move them back to Vn,i. We will do this more precisely below. Let Un,i denote Sn 1(Un,i) and Un,i denote Sn 1(Un,i). − −

The annuli hn 1(Vn,i) Jn,i and Vn,i Jn,i have moduli bigger that ε/dK − \ \ where K is the dilatation of hn 1. Therefore, there exist topological disks −

Ln,i with smooth boundaries and a constant r > 1 satisfying the following properties

Ln,i hn 1(Vn,i) Vn,i • ⊂ − ∩

mod (L J ) r 1 • n,i \ n,i ≥ − mod (Vn,i Ln,i) ε/2dK and mod (hn 1(Vn,i) Ln,i) ε/2dK. • \ ≥ − \ ≥ 47 Now we claim that there exist q.c. maps

χi : hn 1(Un,i), hn 1(Vn,i), Ln,i,Jn,i D5,D3,D2,D1 − − → with uniformly bounded dilatation. That is because all the annuli L J , n,i \ n,i hn 1(Vn,i) Ln,i, and hn 1(Un,i) hn 1(Vn,i) have moduli uniformly bounded − \ − \ − from below and above. The homotopy g : Dom(h) C, for t [0, 1], is t → ∈ deﬁned as

hn 1(z) if z / hn 1(Vn,i) − i − gt(z) :=  ∈ h  1 t ( χi n−1(z) 1)π χi− ( − sin | ◦ |− + 1) χi hn 1(z) if z hn 1(Vn,i). 3 4 ◦ − ∈ −  It is straight to see that g0 hn 1, gt is a well defined homeomorphism for ≡ − every fixed t, and depends continuously on t fro every fixed z. For every

1 2 z ∂Vn,i, at time t = 1, we have g1(z) = χi− ( χi hn 1(z)) ∂Ln,i. That ∈ 3 ◦ − ∈ is, g maps ∂V to ∂L . For the returning part, we consider a q.c. map 1 n,i n,i

Θ : U , V , L ,J D ,D ,D ,D , i n,i n,i n,i n,i → 5 3 2 1 and deﬁne gt+1 : Dom hn 1 C, for t [0, 1], as − → ∈

1 g1(z) if z / g1− ( i Un,i) gt+1(z) :=  ∈  1 t ( χi g1(z) 1)π 1 Θi− ( sin | ◦ |− + 1) Θi g1(z) if z g1− (Un,i). √2 4 ◦ ∈  The homotopy gt for t [0, 2] is the desired adjustment and the map g2 : ∈

Dom hn 1 Range hn 1 is denoted by hn′ 1. − → − −

Let ∆n 1,0 denote the Sn 1-preimage of the domain bounded by the equipo- − − tential of level η(ε) in the dynamic plane of fcn−1 . The multiply connected

domain Ωn 1,0 is deﬁned as −

tn/tn−1

∆n 1,0 Vn,itn−1 . − \ i=0 48 The domains ∆n 1,i and Ωn 1,i, for i = 1, 2,...,tn 1, are deﬁned as the pull − − − i back of ∆n 1,0 and Ωn 1,i, respectively, under f − along the orbit of the critical − − point. Consider the map

˜ 1 hn 1,0 := Sn− 1 hn′ 1 Sn 1 : ∆n 1,0 ∆n 1,0, (3.4) − − ◦ − ◦ − − → − and then, ˜ i i hn 1,i := f − hn 1,0 f : ∆n 1,i ∆n 1,i. (3.5) − ◦ − ◦ − → −

As these maps are compositions of two K1(ε)-q.c., a K 2(ε)-q.c., and possibly some holomorphic maps, they are q.c. with a uniform bound on their dilatation.

By our adjustment in Lemma 3.9 we have hn 1,i maps Ωn 1,i onto Ωn 1,i. − − −

Finally, the annulus Vn 1,0 Wn 1,0, with modulus bigger than ε/2 encloses − \ − Ωn 1,0 and is contained in Vn 1,0. This proves that the domain Ωn 1,0 is well − − −

inside the disk Vn 1,0. Similarly, appropriate preimages of Vn 1,0 Wn 1,0 un- − − \ − i der the conformal maps f − introduce deﬁnite annuli around Ωn 1,i which are − contained in Vn 1,i. In this case, the topological disks Dn,i are the domains Vn,i − which contain the little Julia sets Jn,i well inside themselves.

Case B:

Here, fcn−1 is satellite renormalizable and its second renormalization is of primitive type. Let αn 1 denote the dividing ﬁxed point of fcn−1 , and αn the dividing −

fixed point of it’s first renormalization f − . In this situation, the Julia set R cn 1 of the primitive renormalization f − , and its forward images under f − can R cn 1 cn 1 get arbitrarily close to αn 1 (which is a non-dividing fixed point of fcn−1 ). − R Our idea is to skip the satellite renormalization in this case which essentially imposes the secondary limbs condition on us.

49 Consider an equipotential of level η(ε) contained in Sn 1(Wn 1,0), the ex- − − ternal rays landing at αn 1, and the external rays landing at the fcn−1 -orbit − η(ε) of αn (see Figure 3.3). These rays and the equipotential E depend holomorphically on the parameter within the secondary wake W (η) containing the

tn/tn−1 parameter cn 1. Let us denote fcn−1 by g for simplicity of notation through − this case.

Ray landing at αn 1 −

χ1 Qn 0 A1 0 C0

fcn−1 b

0 Aj 0 Bi

Ray landing at αn

Figure 3.3: Figure of an inﬁnitely renormalizable Julia set. The ﬁrst renormalization is of satellite type and the second one is of primitive type. The puzzle

χ1 piece at the center, Qn , is the ﬁrst puzzle piece in the favorite nest.

Now, using the above rays and equipotential, we introduce some new puzzle

0 pieces. Let Y0 , as before, denote the puzzle piece containing the critical point

and bounded by E(η), the external rays landing at αn 1 and the external rays −

landing at fcn−1 -preimages of αn 1 (i.e. at all ωαn 1 with ω a dth root of unity). − − 0 The External rays landing at αn and their g-preimages, cut the puzzle piece Y0 into ﬁnitely many pieces. Let us denote the one containing the critical point 50 0 0 by C0 , the non-critical ones which have a boundary ray landing at αn by Bi 0 and the rest of them by Aj (these ones have a boundary external ray landing

at some ωαn).

0 The g-preimage of Y0 along the postcritical set is contained in itself. As all processes of making modiﬁed principal nest and the pseudo-conjugacy in Theorem 3.2 are based on pullback arguments, the same ideas are applicable here. The only diﬀerence is that we do not have equipotentials for the second renormalization. As we will see in a moment, certain external rays and part

0 of the equipotential bounding Y0 will play the role of an equipotential for the second renormalization of fcn−1 . By deﬁnition of satellite and primitive renormalizability, every gn(0) belongs to Y 0, for n 0, and there is a ﬁrst moment t with gt(0) A0 (by 0 ≥ ∈ 1 0 t rearranging the indices if required). Pulling back A1 under g along the critical orbit, we obtain a puzzle piece Qχ1 0, such that C0 Qχ1 is a non-degenerate n ∋ 0 \ n 0 annulus. That is because C0 is bounded by the external rays landing at αn

χ1 0 and their g-preimage. Therefore, if Qn intersects ∂C0 at some point on the rays, orbit of this intersection under gk, for k 1, stays on the rays landing at ≥ χ1 0 αn. This implies that image of Qn can not be A1. Also, they can not intersect at equipotentials as they have diﬀerent levels. Now, consider the ﬁrst moment

m χ1 χ1 m m > t when g (0) returns back to Qn , and pullback Qn under g along the

χ1 m critical orbit to obtain Pn . The map g is a unicritical degree d branched

χ1 χ1 covering from Pn onto Qn . This introduces the ﬁrst two pieces in the favorite nest. The rest of the process to form the whole favorite nest is the same as in Section 3.1.2.

tn/tn−1 0 tn/tn−1 0 Consider the map fc −1 : Y fc −1 (Y ), and the corresponding tilde n 0 → n 0 one. One applies Theorem 3.2 to these maps, using the favorite nests intro-

51 duced in the above paragraph, to obtain a q.c. pseudo-conjugacy

tn/tn−1 0 tn/tn−1 0 hn 1 : fc −1 (Y0 ) fc −1 (Y0 ). − n → n The equipotential of level η(ε), the external rays landing at αn 1, and the − external rays landing at the fcn−1 -orbit of αn depend holomorphically on the parameter within the secondary wake W (η) containing the parameter cn 1. −

Therefore, by Proposition 3.3, the dilatation of hn 1 depends on the hyperbolic − distance between cn 1 andc ˜n 1 within one of the ﬁnite secondary wakes W (η) − − under our combinatorial assumption. Thus, it only depends on the a priori bounds ε and the combinatorial condition.

tn/tn−1 j tn/tn−1 j − 0 0 As fc −1 : fcn−1 (Y0 ) fcn−1 (Y0 ), for j = 1, 2,...,tn/tn 1 1, is n → − − univalent, we can lift hn 1 on other puzzle pieces as −

j j tn/tn−1 j 0 tn/tn−1 j 0 hn 1 := fc−−1 hn 1 fc −1 : fc −1 − (Y0 ) fc −1 − (Y0 ) − n ◦ − ◦ n n → n

for these j’s. Since all these maps match the B¨ottcher coordinates, they ﬁt

together to build a q.c. map from a neighborhood of J(fcn−1 ) to a neighborhood of J(fcn−1 ). Further, it can be extended as the identity map in the B¨ottcher coordinates to a q.c. map from the domain bounded by equipotential Eη(ε) to the corresponding tilde domain.

Finally, by Lemma 3.15, we adjust hn 1 to obtain a q.c. map hn′ 1,0 that − − satisﬁes

˜ hn′ 1(Sn 1(Vn+1,itn−1 )) = Sn 1(Vn,itn−1 ), for i =0, 1, 2,...,tn+1/tn 1 1. − − − − − Now, ∆n 1,0 is deﬁned as Sn 1-pullback of the domain bounded by the − − η(ε) equipotential E . The domain Ωn 1,0 is −

tn+1/tn−1 1 −

∆n 1,0 Vn+1,itn−1 . − \ i=0 52 The regions ∆n 1,i and Ωn 1,i, for i =1, 2, 3,...,tn 1, are pullbacks of ∆n 1,0 − − − − i and Ωn 1,0 under f , respectively. Like in the previous case, hn 1,i is deﬁned − − as in Equations (3.4) and (3.5) and satisﬁes

hn 1,i(Ωn 1,i)= Ωn 1,i, for i =1, 2,...,tn 1. − − − − For the same reason as in Case A , Ωn 1,i is well inside the disk Dn 1,i := Vn 1,i. − − −

Case C :

Here, fcn−1 is twice satellite renormalizable. The argument in this case relies more on the compactness of the parameters under consideration rather than a dynamical discussion.

Figure 3.4: A twice satellite renormalizable Julia set drawn in grey. The dark part is the Julia Bouquet B2,0.

53 Little Julia sets J1,i of the ﬁrst renormalization of fcn−1 touch at the dividing

ﬁxed point αn 1 of fcn−1 . Note that αn 1 is one of the non-dividing ﬁxed − −

points of the ﬁrst renormalization of fcn−1 . The union of these little Julia sets is called Julia bouquet and denoted by B1,0. Similarly, the little Julia sets

J2,i, for i = 0, 1, 2,..., (tn+1/tn 1) 1, of the second renormalization of fcn−1 − − are organized in pairwise disjoint bouquets, B2,j, touching at a periodic point.

That is, each B2,j consists of tn+1/tn little Julia sets J2,i touching at one of

their non-dividing ﬁxed points. As usual, B2,0 denotes the bouquet containing the critical point. See Figure 3.4.

By an equipotential of level η(ε) (which encloses Sn 1(Wn 1,0)) and the − −

external rays landing at αn 1, we form the puzzle pieces of level zero. Recall − 0 that Y0 denotes the one containing the critical point. The following lemma states that the bouquets are well apart form each other.

Lemma 3.10. For parameters in a ﬁnite number of truncated secondary limbs, modulus of the annulus Y 0 B is uniformly bounded above and below. 0 \ 2,0

To prove this lemma we first need to recall some definitions. Let X and Y be compact subsets of C equipped with the Euclidean metric d. The Hausdorff distance between X and Y is defined as

dH (X,Y ) := inf ε : Y Bε(X), and X Bε(Y ) . ε { ⊂ ⊂ }

The space of all compact subsets of C endowed with this metric is a complete metric space. A set valued map c X , with X compact in C, is upper semi-continuous → c c if c c implies that X contains the Hausdorff limit of X . n → c cn A family of simply connected domains Uλ depends continuously on λ if there exit choices of uniformizations ψ : D U continuous in both variables. λ 1 → λ 54 A family of polynomial-like maps (P : V U ,λ Λ), parametrized on λ λ → λ ∈ a topological disk Λ, depends continuously on λ if Uλ is a continuous family of simply connected domains in C, and for every fixed z C, P (z) depends ∈ λ continuously on λ where ever it is defined.

Proposition 3.11. Let (P : V U ,λ Λ) be a continuous family of λ λ → λ ∈ polynomial like maps with connected ﬁlled Julia sets K . The map λ K is λ → λ upper semi-continuous.

Proof. Assume that λ λ, K := K(P ), and K := K(P ). To prove that n → λn λn λ λ

K := lim Kλn is contained in Kλ, it is enough to show that for every ε > 0, K B (K ) for suﬃciently large n. λn ⊆ ε λ To see this, assume that z / B (K ). If z / V , then by continuous ∈ ε λ ∈ λ dependence of V on λ, z / K for large n. If z V , then there exists a λ ∈ λn ∈ λ positive integer l with P l (z) U V . As P : V U converges to λ ∈ λ \ λ λn λn → λn P : V U , P l (z) or P l+1(z) belongs to U V . Therefore, z / K . λ λ → λ n n λn \ λn ∈ λn

Proof of Lemma 3.10. Let cn 1 be a twice satellite renormalizable parameter − in a truncated secondary limb. Consider the external rays landing at the

0 dividing ﬁxed point α of f − and it’s preimages under f − . Let X n R cn 1 R cn 1 0 denote the puzzle piece obtained from these rays which contains the critical

2 0 point. As f − is twice satellite renormalizable, f − : X C is a branched cn 1 R cn 1 0 → 0 covering map over its image. One can consider a continuous thickening of X0 , described in Section 3.1.2, to form a continuous family of polynomial-like maps parametrized over this truncated limb. The little Julia set of this map was

denoted by J2,0 and the Julia bouquet B2,0 is the connected component of

tn+1/tn−1 1 −

fcn−1 (J2,0) i=0 containing the critical point. 55 For cn 1 in a ﬁnite number of truncated secondary limbs, the Julia bouquet −

B2,0 is unoin of a ﬁnite number of little Julia sets. Hence, by above lemma, it

depends upper semi-continuously on cn 1. As B2,0 is contained well inside the − 0 interior of Y0 for such cn 1 which belongs to a compact set of parameters, we − conclude that modulus of Y 0 B is uniformly bounded below. 0 \ 2,j To see that this modulus is uniformly bounded above, one only needs to

observe that αn and 0 belong to B2,0 and are strictly apart from each other for these parameters.

By Lemma 3.10 there are simply connected domains Ln′ 1 Ln 1 and − ⊆ −

Ln′ 1 Ln 1 such that moduli of the annuli − ⊆ −

0 Y0 Ln 1, Ln 1 Ln′ 1, Ln′ 1 B2,0 \ − − \ − − \ 0 Y0 Ln 1, Ln 1 Ln′ 1, Ln′ 1 B2,0 \ − − \ − − \ are bigger than some constant depending only on the combinatorial class . SL Moreover, the ratios

0 mod (Y0 Ln 1) mod (Ln 1 Ln′ 1) mod (Ln′ 1 B2,0) \ − , − \ − , − \ 0 mod (Y0 Ln 1) mod (Ln 1 Ln′ 1) mod (Ln′ 1 B2,0) \ − − \ − − \ are also uniformly bounded below and above independent of n. Theorem 2.9, combined with above data, implies that there exits a q.c. map 0 ˜ 0 hn 1 : Y0 Ln 1 Y0 Ln 1, with a uniform bound on its dilatation, which − \ − → \ − 0 matches the B¨ottcher marking on the boundary of Y0 . We further lift hn 1 − i i via fc−−1 and fc˜−−1 to extend hn 1 to q.c. maps n n −

i 0 ˜ i ˜ 0 ˜ tn hn 1 : fc−n−1 (Y0 Ln 1) fc−n−1 (Y0 Ln 1), for i =1, 2,..., 1 − \ − → \ − tn 1 − − with the same dilatation. The domain of each such map is a puzzle piece

0 l Yj (with j = tn/tn 1 i) cut off by the equipotential of level η/d and a − − i component of fc−−1 (∂Ln 1). As all these maps match the Böttcher marking on n − 56 0 the boundaries of Yj , they can be glued together. Finally, one extends this map as the identity in the Böttcher coordinates onto spaces between equipotential of level η and equipotentials of level η/dl. we denote this extended map with

the same notation hn 1. As it is lifted under and extended by holomorphic − maps, there is a uniform bound on its dilatation.

Let ∆n 1,0 be the Sn 1-preimage of the domain inside equipotential E(η) − − i and Ln 1,i be the component of the Sn 1-preimage of fc−−1 (Ln 1)) enclosing − − n −

the little Julia set Jn 1,i. We deﬁne the multiply connected regions −

tn/tn−1 1 −

Ωn 1,0 := ∆n 1,0 Ln 1,itn−1 − − \ − i=0 i Like before, ∆n 1,i and Ωn 1,i are deﬁned as f -preimage of ∆n 1,0 and Ωn 1,0 − − − −

containing or enclosing Jn 1,i, respectively. − We have

1 hn 1,0 := Sn− 1 hn 1 Sn 1 : ∆n 1,0 ∆n 1,0, with hn 1,0(Ωn 1,0)= Ωn 1,0. − − ◦ − ◦ − − → − − − −

Also,

i i hn 1,i := f − hn 1,0 f : ∆n 1,i ∆n 1,i, with hn 1,i(Ωn 1,i)= Ωn 1,i. − ◦ − ◦ − → − − − − As the equipotential η(ε) is contained in Sn 1(Wn 1,0), Ωn 1,0 is contained − − −

in Wn 1,0. Therefore, Ωn 1,0 is well inside Vn 1,0. Conformal invariance of − − −

modulus implies that the other domains Ωn 1,i are also well inside Vn 1,i. This − − completes the construction in Case C .

To fit together the multiply connected domains Ωn(k),i and the q.c. maps h : Ω Ω , we follow the word of cases introduced at the be- n(k),i n(k),i → n(k),i ginning of the construction. In Cases A and B, we have adjusted hn 1 such − that it sends ∂Vn 1,0 to ∂Vñ 1,0. Therefore, if any of the three cases follows − − Case A or B, we consider nf : W V and straighten it with these R n,0 → n,0 57 choices of domains (instead of nf : V U ). If a case of construction R n,0 → n,0 on level n is following Case C , the set ∆n,0 introduced on level n is replaced by ∆n,0 Sn(Ln′ 1), hn,0 is restricted to this set, and adjusted so that it sends ∩ − ˜ ˜ Sn(Ln′ 1,i) to Sn(Ln′ 1,i). The annulus Ln 1 Ln′ 1 provides a definite annulus − − − \ − separating Ωn,0 and Ln 1,0. −

In the following two sections, we will denote the holes of Ωn,i by Vn+1,j, A B 1 that is, Vn+1,j is Vn+1,j, if n belongs to Case or or Vn+1,j is Sn− (Ln,j) if n belongs to Case C .

i 3.2.4 The gluing maps gn(k)

In this section we build K′(ε)-q.c. maps

gi : V ∆ V ∆ . n(k) n(k),i \ n(k),i → n(k),i \ n(k),i i Every gn(k) has to be identical with hn(k),i on ∂Vn(k),i and with hn(k+1),i on i ∂∆n(k),i (which is outer boundary of Ωn(k),i). Then gluing all these maps gn(k)

and hn(k),i produces a q.c. map H with dilatation bounded by maximum of K

and K′(ε). In what follows, for simplicity of notation, we use index n instead of n(k), and assume that n runs over subsequence n(k). So for all n, means for all n(k)’s.

0 Like previous steps, it is enough to build gn for all n, and pull them back i i by f − to obtain gn. Properties of the maps hn,i and the domains Vn,i and

Ωn,i implies that these maps glue together on the boundaries of their domain’s

of deﬁnition. Again, we drop the superscript index if it is 0, i.e. gn denotes

0 the map gn. To build a q.c. map from an annulus to another annulus with given boundary conditions, there is a choice in the number of “twists” one may make. Moreover, to have a uniform bound on the dilatation of such a map, the two annuli must have proportional moduli bounded below, and the 58 number of twists has to be uniformly bounded. Note that the number of twists eﬀects on the homotopy class of the ﬁnal map H. In this section, we show that the corresponding annuli V ∆ and n,0 \ n,0 V ∆ have proportional moduli with a constant depending only on ε. In n,0 \ n,0 the next section we specify the number of twists needed for the isotopy class of a Thurston conjugacy.

Lemma 3.12. Let U ′, U, and U be three annuli whose inner boundaries are

D such that mod (U U ′) ε, and mod (U) M, for some constant M. 1 \ ≥ ≤ Let ψ be a K-q.c. map from U onto U. For every r with D U ′ ψ(U ′), r ⊂ ∩ moduli of U D and U D are proportional with a constant depending only \ r \ r on M, K, and ε. Proof. By properties of q.c. maps we have

ε mod (U D ) M, ≤ \ r ≤ ε mod (U D ) KM K ≤ \ r ≤ which implies the lemma.

Lemma 3.13. Moduli of the annuli V ∆ and V ∆ are proportional n,0 \ n,0 n,0 \ n,0 with a constant depending only on ε. Proof. If level n follows Case A or B, one proves this by applying the above

tn lemma to images of Vn,0, Wn,0 = f − (Vn,0) and ∆n,0 under Bc( nf) Sn and the R ◦

n corresponding tilde objects, where Bc( f) is the B¨ottcher coordinate for Pcn . R

Note that ∆n,0 is mapped to Dr. To obtain an upper bound M, it is enough to go several levels lower than the fundamental annulus to get an annulus with bounded modulus.

If level n follows Case C , by deﬁnition, V ∆ is L L′ and V˜ ∆˜ n,0 \ n,0 n \ n n,0 \ n,0 is L˜ L˜′ which were chosen to be proportional. n \ n 59 Given a curve ℓ U which is given as γ :[a, b] U with γ(a) on the inner ⊂ → boundary of U (corresponding to the bounded component of C U) and γ(b) \ on the outer boundary of U (corresponding to the unbounded one), deﬁne the wrapping number ω(ℓ) as

1 b ∂π (φ(γ(t))) 2 dt. 2π ∂θ a where φ is a uniformization of the annulus U by a round annulus, and π2(z) is the polar angle of the point z. Basically, ω(ℓ) is total turning of the curve ℓ in the uniformized coordinate. Note that ω(ℓ) is invariant under the automor- phism group of U. So, it is independent of the choice of uniformization and just like winding number, it is constant over the homotopy class of all curves with same boundary points. Let A(r) denote the round annulus D D . r \ 1

Proposition 3.14. Given ﬁxed constants K 1 and r > 1, there exists a ≥ constant N such that, for every K-q.c. map ψ : A(r) A(r′), the wrapping → number of the curve ψ(t), t [1,r], belongs to the interval [ N, N]. ∈ −

Proof. Consider the curve family L consisting of all ray segments in A(r) obtained from rotating the real line segment [1,r] about the origin, that is, the radial lines in A(r). We denote by Γ(F ) the extremal length of a given curve family F . The inequality Γ(L)/K Γ(ψ(L)) KΓ(L) holds for the K-q.c. map ψ. See [Ah66] ≤ ≤ for more details on curve families and extremal length properties. It is easy to see that if the interval [1,r] is mapped to a curve with wrapping number T , then every curve in L is mapped to a curve with wrapping number between T + 1 and T 1. − By deﬁnition of extremal length and choosing conformal metric ρ as the

60 Euclidean metric, we obtain

2 2 inf ℓρ(ψ(γ)) 4π(T 1) KΓ(L) Γ(ψ(L)) = sup 2 − , ≥ ρ S ≥ R 1 ρ ′ − K and Γ(L) = log R/2π. By properties of q.c. mappings, we have R′ R which ≤ implies 1 K log(R)(R2K 1) T − +1 ≤ π 8

Lemma 3.15. Fix round annuli A(r), A(r′), a positive constant K1, and an

integer k with mod A(r′)/K mod A(r) K mod A(r′) and mod A(r) 1 ≤ ≤ 1 ≥ δ. If homeomorphisms h : ∂D ∂D ′ and h : ∂D ∂D have K -q.c. 1 r → r 2 1 → 1 2 extensions to some neighborhood of these circles for some K2, then there exists a K-q.c. map h : A(r) A(r′) such that →

h(z)= h (z) for every z ∂D , and h(z)= h (z) for every z ∂D • 1 ∈ r 2 ∈ 1

The curve h(t), for t [1,r], has wrapping number θ(h (r)) θ(h (1))+k. • ∈ 1 − 2

Moreover, K depends only on K1, K2, k and δ.

Proof. One proves this statement by explicitly building such maps for every k. More details left to reader.

Applying the above lemma to the uniformization of the annuli V ∆ n,0 \ n,0 and Vn,0 ∆n,0, the induced maps from hn 1,0 and hn,0 on their boundaries \ − and a number k , which is speciﬁed later, gives the required K -q.c. maps g . n ′ n

In the next section we specify some especial numbers kn ,which are bounded by a constant depending on ε, in order to make the K-q.c. map H, obtained after gluing all these maps, homotopic to a topological conjugacy relative the postcritical set.

61 Deﬁnite moduli of the annuli V ∆ implies that the holes V shrink to n,i \ n,i n,i points in the postcritical set. Therefore, H can be extended to a well deﬁned K-q.c. map on the postcritical set. See [St55] for a detailed proof of this and further results on quasi-conformal removability.

3.2.5 Isotopy

Denote by ψn the topological conjugacy between fcn and fc˜n obtained from extending the identity map in the B¨ottcher coordinates onto Julia set. The lift 1 ψ = S˜ − ψ S is a topological conjugacy between nf and nf˜ on a n,0 n ◦ n ◦ n R R neighborhood of the little Julia set Jn,0. Note that this neighborhood covers the domain Ωn,0. In the dynamic plane of fcn , let U(η) denote the domain enclosed by the equipotential of level η(ε).

Lemma 3.16. If level n belongs to one of Case A or B, then the q.c. map h : ∆ ∆ , for i =0, 1,...t 1, is homotopic to n,i n,i → n,i n −

i 1 i ψ := f − S˜− ψ S f : ∆ C n,i ◦ n ◦ n ◦ n ◦ n,i → relative the little Julia sets Jn+1,j of level n +1 inside ∆n,i.

˜ Note that ψn,i(∆n,i) is a neighborhood of the little Julia sets Jn+1,i+tnj contained in ∆n,i.

Proof. First assume that level n follows a Case A or B. From the deﬁnition of the domains ∆n,i, Vn,i and the q.c. maps hn,i, it is enough to prove the statement for i = 0. For other ones one pulls the homotopies back by f i, or constructs them in the same way as for i = 0).

As ∆n,0, ψn,0, and the q.c. map hn,0 are lifts of ∆n,0, ψn, and hn,′ 0 by the straightening map, it is enough to make the homotopy on the dynamic planes

62 of f and f and then transfer it to the dynamic planes of nf and nf˜ by cn c˜n R R

the straightening map. Recall that in our construction, hn,′ 0 is an adjustment

of hn,0 through a homotopy relative the little Julia sets of the next level. Thus,

to prove the lemma, we only need to prove that hn,0 and ψn are homotopic relative the little Julia sets. Assume that level n belongs to Case A . The idea of the proof, presented in

detail below, is to divide the domain ∆n,0, by means of rays and equipotential

arcs, into some topological disks and one annulus such that ψn and hn,0 are identical on the boundaries of these domains.

χn χn qχn χn Consider the puzzle piece Qn,0 (Qn,0 = Y0 ). The equipotential fc−n (E(η)),

χn χn and the rays of the puzzles Qn,i up to equipotential fc−n (E(η)), cut the domain

χn ∆ into one annulus ∆ f − (U(η)) and some topological disks. The topo- n,0 n,0 \ logical disks which do not intersect the little Julia sets of level n, the puzzle

χn χn pieces Q , and the remaining annulus E(η) f − (E(η)), are the appropriate n,i \

domains. By Theorem 3.2 the maps hn,0 and ψn are identical in the B¨ottcher coordinate on the boundaries of these domains. Indeed, the topological con-

jugacy ψn between fcn and fc˜n is identity in the B¨ottcher coordinate and the

pseudo-conjugacy hn,0 obtained in Theorem 3.2 matches the B¨ottcher marking.

χn This proves that the two maps are homotopic outside of the puzzle pieces Qn,i.

χn To ﬁnd a homotopy inside the pieces Qn,0, recall that we started with a q.c. map on Qχn P χn , which was homotopic to ψ relative the boundaries. n,0 \ n,0 n So all the pullbacks of this map on the annuli Ak Ak+1 are homotopic to ψ n \ n n relative boundaries. As the two maps are identical on the little Julia sets, we conclude that the two maps are homotopic relative the little Julia sets. If level n belongs to Case B, we repeat above argument on puzzle pieces of level zero.

63 The same proof works if level n follows Case C . The only diﬀerence is that the domain of deﬁnition of the homeomorphisms are restricted to a smaller set. Thus, we may restrict the homotopy onto that set.

In the following paragraphs we assign the number of twists kn for the gluing

maps gn. If level n belongs to Case A or B, and it follows a Case A or B, consider the uniformizations φ : A(s) (V K ), φ : A(r) (∆ K ), 1 → n,0 \ n,0 2 → n,0 \ n,0 φ : A(˜s) (V K˜ ) and φ : A(˜r) (∆ K˜ ) by round annuli. 1 → n,0 \ n,0 2 → n,0 \ n,0 The q.c. maps hn 1,0 and hn,0 lift via φi and φi to q.c. maps hˆn 1,0 : A(s) − − → V (˜s) and hˆ : A(r) V (˜r) with the same dilatation. By composing the n,0 → uniformizations with rotations, we may assume that the point one is mapped to the point one by these two maps. By Proposition 3.14, the image of the ˆ segment [1,s] under the q.c. map hn 1 has wrapping number ω1n bounded by −

some N and image of the segment [1,r] under the q.c. map hˆn,0 has wrapping number ω bounded by N which depends only on ε. Deﬁne k as ω ω , 2n n 1n − 2n ˆ ˆ and note that gluing hn,0 and hn 1,0 in Lemma 3.15 by such a choice makes −

the image of the segment [1,s] under the two maps gn and hˆn,0 glued together, ˆ homotopic to image of the segment [1,s] under hn 1,0 relative two boundary −

circles. This homotopy lifts to a homotopy between hn 1,0 and the two maps −

gn and hn,0 glued together.

Before we deﬁne kn for the other cases, we need to show that the q.c. map C hn′ 1, built in Case , has a q.c. extension over the topological disk Ln 1. − − C Lemma 3.17. The q.c. map hn′ 1 introduced in Case has a q.c. extension −

onto Ln 1 with a uniform bound on its dilatation depending only on ε. More- −

over, this extension can be made homotopic to ψn 1 relative the Julia bouquet −

B2,0.

64 Proof. Consider the fundamental annuli Sn 1(Un,0 Vn,0) and Sñ 1(Uñ,0 Vñ,0) − \ − \

for the ﬁrst renormalizations of fcn−1 and fc˜n−1 . Let

gn : Sn 1(Un,0 Vn,0) Sñ 1(Uñ,0 Vñ,0) − \ → − \ be a q.c. map which satisfies the equivariance relation on the boundaries of these annuli. By lifting gn on the preimages of these annuli we obtain a q.c. map gn from complement of the little Julia set J1,0 to the complement of the ˜ little Julia set J1,0 on the dynamic planes of fcn−1 and fcñ−1 . By Lemma 3.8, gn (or some rotation of it) can be extended as ψn 1 onto the Julia set J1,0. − Moreover these two maps are homotopic relative this little Julia set. We then adjust gn so that it maps Ln′ 1 to Ln′ 1. − − 0 Consider the three annuli Y0 Ln 1, Ln 1 Ln′ 1, and Ln′ 1 B2,0, as well \ − − \ − − \ 0 0 as the corresponding tilde ones. We have hn 1 : Y0 Ln 1 Y0 Ln 1 − \ − → \ − and gn : Ln′ 1 B2,0 Ln′ 1 B2,0. To glue these two maps on the middle − \ → − \ annulus, we use the above argument to find the right number of twists on this annulus. Consider a curve γ connecting a point a on the bouquet B2,0 to a

0 point d on the boundary of Y0 such that it intersects the boundaries of Ln′ 1 −

and Ln 1 only once which are denoted by b and c. Lets denote by γab, γbc, and −

γcd each segment of this curve cut oﬀ by these four points. The real number

ω(ψ(γ)) ω(hn 1(γcd)) ω(gn(γab)) is uniformly bounded, by Proposition 3.14, − − − depending only on ε. If we glue hn 1 and gn by such a number of twists (see − 0 Lemma 3.15), the resulting map will be homotopic to ψn 1 relative B2,0 ∂Y0 . − ∪ 0 Note that the two maps hn′ 1 and ψn 1 are identical on the boundary of Y0 . − −

Thus, one can extend this map over the other topological disks Ln 1,i. We will −

denote the ﬁnal q.c. map by hn′ 1. −

If a Case C follows a Case A or B, the number of twists kn is defined as the one introduced above. If level n 1 belongs to Case C and level n is − 65 any of the three cases, we use the uniformization of the annuli V B and n,0 \ n,0 ∆ B , as well as the corresponding tilde ones, instead of the above annuli, n,0 \ n,0 to define kn. The following lemma is frequently used in the final proof of isotopy. Its prove is left to reader.

Lemma 3.18. Let U and U be closed annuli with ∂U = γ γ , and ∂U = 1 ∪ 2 γ γ . Also, let ht : γ γ˜ , for t [0, 1], i = 1, 2, be two continuous 1 ∪ 2 i i → i ∈ families of homeomorphisms, and the homeomorphism G0 : U U be an → interpolation of h0 and h0 on U. Then G0 can be extended to a continuous 1 2 family of interpolations of ht and ht , for t [0, 1]. 1 2 ∈

i Proposition 3.19. The K-q.c. map H obtained from gluing all the maps gn and hn,k is homotopic to the topological conjugacy Ψ relative the postcritical set (f). PC

0 k l Proof. Let Hn denote the map obtained from gluing the maps g1,g ,...,gn 1 − and h1,0, h2,i,...,hn,j, for all possible indices i, j, k and l up to level n.

First, we show that H1 is homotopic to the topological conjugacy Ψ between fc and fc˜ relative the little Julia sets J1,i. Then we show that each Hn 1 is − homotopic to Hn relative the little Julia sets of level n + 1.

The map H1 is just h1,0 which is homotopic to ψ1,0, by Lemma 3.16 or Lemma 3.17. It is homotopic to Ψ by Lemma 3.8.

Recall that the two maps Hn 1 and Hn are identical on the complement of −

Vn,j. Inside ∆n,0, Hn 1 and Hn are hn 1,0 and hn,0, respectively. − − The domain V is divided into annulus V ∆ and the topological n,0 n,0 \ n,0 disk ∆n,0. On∆n,0, hn,0 and hn 1,0 are homotopic to ψn,0 relative iJn+1,i by − ∪ t Lemmas 3.16 and 3.17. Thus, there exists a homotopy hn, for t in [0, 1], which starts with hn,0 and ends with hn 1,0, such that it maps ∂∆n,0 to ∂∆n,0, for all − 66 t [0, 1]. At time zero consider the map h on the inner boundary of this ∈ n,0 annulus, hn 1,0 on the outer boundary of this annulus and the interpolation − 0 0 Gn = gn between them. Applying above lemma to the ﬁxed homeomorphism t hn 1,0 on the outer boundary, and hn on the inner boundary, we obtain a − t 1 continuous family of interpolations Gn between them. The map Gn is a homeomorphism from the annulus to itself which is an interpolation of hn 1,0 on the − boundaries, but this interpolation has to be homotopic to hn 1,0 on the annu- − lus. Indeed, these two maps send a curve joining the two diﬀerent boundaries to two curves (joining the two boundaries) which are homotopic relative end points. This comes from our choice of wrapping numbers for the gluing maps. Let t = 0 < t < t < < 1, be an increasing sequence of numbers 0 1 2 t in [0, 1]. Assume that H , for t in [t0, t1], denotes the homotopy obtained

t above between Ψ and H1 relative the little Julia sets J1,i. Also, let H , for t [t , t ], and n = 1, 2,... , denote the homotopy between H and H ∈ n n+1 n n+1 relative the little Julia sets of level n + 2. It is clear from the construction that Ht(z) eventually stabilizes for any ﬁxed z, and equals to H(z). Indeed, a priori bounds assumption implies that the diameter of the topological disks V tends to zero as n . Therefore, n,i →∞ the uniform distance between Ht and H is going to zero as t 1. We conclude → that Ht for t in [0, 1] is a homotopy between Ψ and the Thurston conjugacy H relative the postcritical set.

3.2.6 Promotion to hybrid conjugacy

Proposition 3.20. Suppose all inﬁnitely renormalizable unicritical polynomials in a given combinatorial class τ = , , ,... , satisfying con- {M1 M2 M3 } SL dition, enjoy a priori bounds. Then q.c. conjugacy implies hybrid conjugacy

67 for maps in this class.

Proof. Assume that this is not correct and there are polynomials P1 and P2 in this class which are q.c. equivalent but not hybrid equivalent. Deﬁne the set

Ω= c C : P is q.c. equivalent to P { ∈ c 1} = c C : P is q.c. equivalent to P . { ∈ c 2} We show that the set Ω is both open and closed in C which is not possible. Theorem 3.4 implies that q.c. conjugacy is equivalent to combinatorial conjugacy for maps in this class. Since every combinatorial class is an intersection of closed sets (connectedness locus copies), Ω is closed. Consider a point P in Ω. The polynomial P is not hybrid equivalent to both of P1 and P2 by assumption. Assume that it is not hybrid equivalent to P1 (for the other case just change P to P ). Let φ : C C, K-q.c. conjugates P 1 2 1 → to P , i.e. φ P = P φ . By pulling back the standard complex structure 1 1 ◦ 1 ◦ 1 0 on C under φ1, we obtain a complex structure on C with dilatation bounded

K 1 by − . Consider the complex structures := λ , for λ in the disk of radius K+1 λ K+1 K 1 around origin. − By applying measurable Riemann mapping Theorem 2.8 , there are q.c. maps φ which map complex structure to , and leave the points 0 and λ λ 0 ∞ 1 K+1 ﬁxed. The maps Pλ := φλ− P1 φλ, for λ in the disk of radius K 1 around ◦ ◦ − origin, are holomorphic maps of the same degree as the degree of P1, and send

inﬁnity to inﬁnity with the same local degree as the one of P1. Thus, they are polynomials. For λ = 1 we obtain the polynomial P , and for λ = 0 we

obtain P1. By analytic dependence of the solution of the measurable Riemann

mapping theorem on the complex structure, Pλ cover a neighborhood of P in Ω. This shows that P is an interior point in Ω. As P was an arbitrary point in Ω, we conclude that Ω is open.

68 3.2.7 Dynamical description of the combinatorics

In this section we give a dynamical definition of some combinatorial classes. It has been proved in [KL05] that infinitely renormalizable parameters satisfying decorations enjoy a priori bounds. Let c be an infinitely renormalizable parameter with renormalizations f := n(P ), their straightening f , and n R c cn 1 critical puzzle piece of level 1 denoted by Y0 (n). The dynamical meaning of a parameter c satisfying the decoration condition is that there exists a constant M such that for every n 0 there are integers t and q , both bounded by M, ≥ n n with

f kqn (0) Y 1(n), for every k < t , and every n 1, • n ∈ 0 n ≥

f tnqn (0) / Y 1, for every n 1. • n ∈ 0 ≥ In particular, this condition implies that the number of rays landing at the dividing fixed point of fn (here qn) is uniformly bounded. An infinitely renormalizable parameter is of bounded type if the relative return times tn+1/tn of the renormalizations n(f) = f tn are bounded by some constant M. It fol- R lows from definition that the decoration condition includes infinitely primitive renormalizable parameters of bounded type. In Section 3.1.3, we associated a sequence of maximal connectedness locus copies τ(f) = 1, 2,... to every infinitely renormalizable unicritical M M polynomial-like map f. Let π (τ(f)) = n denote the projection map onto n M n’s component. Define

c is at least n times renormalizable, and τ(f, n)= c d . ∈M πi(τ(f)) = πi(τ(Pc)), for i =1, 2,...,n In other words, the connectedness locus copy τ(f, n) is the set of at least n times renormalizable parameters with the same combinatorics as of f up to 69 level n. Given an inﬁnitely renormalizable map f, and an increasing sequence of positive integers n , we deﬁne a sequence of relative connectedness locus i copies of as follows: Md

(˜τ(f), n ) := ˜ n1 , ˜ n2 ,..., ˜ nk ,... , { i} M M M

where,

nk nk−1 ˜ := τ( f, nk nk 1). M R − −

One can see that there is a one to one correspondence between the two sequences τ(f) and (˜τ(f), n ) for every increasing sequence n . Thus, one { i} i may take the latter one as definition of the combinatorics of f. Consider the main hyperbolic component of the connectedness locus . Md There are infinitely many primary hyperbolic components attached to this component (corresponding to rational numbers). Similarly, there are infinitely many hyperbolic components, secondary ones, attached to these primary components, and so on. Consider the set of all hyperbolic components obtained this way. That is, the ones which can be connected to the main hyperbolic component by a chain of hyperbolic components bifurcating one from another. The closure of this set after adding all possible components of its complement

is called the molecule Md. We say that an inﬁnitely renormalizable map f satisﬁes the molecule condition, if there exists a positive constant η > 0 and an increasing sequence of

ni positive integers n ∞ , such that for every i 1, f is a primitive renormal- ii=1 ≥ R ni 1 ni ization of − f, and moreover, the Euclidean distance between ˜ and the R Md molecule is at least η. Note that for a map satisfying this condition, there Md may be infinitely many satellite renormalizable maps in the sequence nf . {R } 70 However, the condition requires that there are infinite number of primitive levels with the corresponding relative connectedness locus copies uniformly away from the molecule. One can see that the parameters in the decoration condition satisfy the molecule condition. For every ε 0, and every hyperbolic component of the connectedness ≥ locus, there are at most finite number of limbs attached to this hyperbolic component with diameter bigger than ε (see [H93]). This implies that for every η > 0, all the secondary limbs except finite number of them are contained in the η neighborhood of the molecule. This implies that the parameters satisfying the molecule condition also satisfy condition, Therefore, we have the following. SL

Corollary 3.21. Let f and f˜ be two inﬁnitely renormalizable quadratic polynomials satisfying the molecule condition. If f and f˜ are combinatorially equivalent, then they are conformally equivalent.

71 Chapter 4

Typical Trajectories of complex quadratic polynomials

4.1 Accumulation on the ﬁxed point

4.1.1 Post-critical set as an attractor

Let f : U Cˆ Cˆ be a holomorphic map. Given a point z U, if f(z) ⊆ → ∈ belongs to U we can define f 2(z) = f f(z). Similarly, if f 2(z) also belongs ◦ to U, f 3(z) is defined and so on. Orbit of z, denoted by (z), is the sequence, O z, f(z), f 2(z),... , as long as it is defined. So it might be a finite or an infinite sequence. We say that (z) eventually stays in a given set E Cˆ, if there O ⊂ exists an integer k such that f i(z) E, for every integer i k. ∈ ≥ Distortion of a map f : U Cˆ Cˆ is the supremum of log( f ′(z)/f ′(w) ) ⊆ → | | (in the spherical distance) for all z and w in U, which might be finite or infinite. We say that a simply connected domain U C, different from C, has ⊂ bounded eccentricity, if there exists a univalent onto map ψ : B(0, 1) U → (a uniformization) with bounded distortion. One can see that if a simply

72 connected domain U has bounded distortion M, then ratio of radii of smallest circle containing U, and largest circle contained in U, is less than some constant depending only on M. We frequently use the following distortion theorem due to Koebe [P75] to transfer areas under holomorphic maps.

Theorem 4.1. (Koebe distortion theorem) Suppose that f : B(0, 1) C is a → univalent function with f(0) = 0, and f ′(0) = 1. For every z B(0, 1) we ∈ have the following estimates

z z | | | | (1) (1+ z )2 f(z) (1 z )2 , | | ≤| | ≤ −| |

1 z 1+ z −| | | | (2) (1+ z )3 f ′(z) (1 z )3 , | | ≤| | ≤ −| |

1 z 1+ z −| | | | (3) 1+ z zf ′(z)/f(z) 1 z . | | ≤| | ≤ −| | This implies the 1/4 theorem: The domain f(B(0, 1)) contains B(0, 1/4).

The following result in [Ly83b] shows that the post-critical set of a rational map attracts orbit of almost every point in the Julia set.

Proposition 4.2. Let f : Cˆ Cˆ be a rational map with J = Cˆ and V be an → arbitrary neighborhood of (f). Then, the orbit of almost every point in the PC Julia set of f eventually stays in V .

Here, we give a simple argument based on the Montel’s normal family theorem for readers convenience. Also, we will use this approach in the next section, with iterating a renormalization instead of the map itself, to examine the Lebesgue measure of certain post-critical sets.

Proof. For a given domain V (f), deﬁne the set ⊃ PC

Γ := z J for inﬁnitely many integer k > 0, f k(z) / V . { ∈ | ∈ } 73 If area of Γ is not zero, let z be a Lebesgue density point of Γ. Let nk

be an increasing sequence of positive integers with f nk (z) / V , and let y be ∈ an accumulation point of the sequence f nk (z) . As y / V , it has a def- ∈ inite distance δ from (f). For suﬃciently large n , let E denote the PC k nk n component of f − k (B(y,δ/2)) containing z. As B(y,δ/2) does not intersect

(f), f nk : E B(y,δ/2) is univalent, and in addition, its inverse has a PC nk → univalent extension over the larger domain B(y, δ). By the Koebe distor-

tion theorem, all the domains Enk have bounded eccentricity, and the maps f nk : E B(y,δ/2) have uniformly bounded distortion. nk → If E ’s do not shrink to z as n , their uniformly bounded eccentricity nk k →∞ implies that Enk ’s contain a ball B(z,r), for some constant r > 0. Thus, every member of the sequence f nk maps B(z,r) into B(y, δ). This implies that f nk is a normal family by Montel’s theorem, contradicting the choice of z in { } the Julia set. Therefore, diameter of Enk tends to 0.

The family Enk shrink regularly to z, i.e. there exists a constant c> 0 such that for each Enk , there exists a round ball B with

E B, and area(E ) c area B. nk ⊂ nk ≥

As z is a Lebesgue density point of Γ (and so of J), Lebesgue’s Theorem implies that, area(E Γ) lim nk =1. nk area(E ) →∞ nk As f nk ’s have bounded distortion, and Γ is f invariant, we have

nk area(f (Enk Γ)) area(B(y,δ/2) Γ) lim n = lim =1. nk area(f k (E )) nk area(B(y,δ/2)) →∞ nk →∞ One concludes from the last equality, and that Γ J, to get B(y,δ/2) J. ⊆ ⊆ This implies that J = Cˆ, contradicting our assumption.

74 4.1.2 The Inou-Shishikura class and the near-parabolic renormalization

Continued fractions: We use a slightly diﬀerent type of continued fractions deﬁned as follows. Any irrational number α R Q can be written ∈ \ as an accelerated continued fraction of the form:

ε0 α = a0 + ε1 a1 + ε2 a + 2 . .. where a Z and ε = 1, for n = 0, 1, 2,.... For any real number α R, n ∈ n ± ∈ deﬁne α := min x n : n Z . Let α = α and a be the closest integer {| − | ∈ } 0 0 1 to α, so that α = a0 α0. For every n 0, let αn+1 = and an+1 be the ± ≥ αn closest integer to 1 . Then the signs ε are determined by 1 = a + ε α . αn n αn−1 n n n Note that α belongs to (0, 1/2) for every n 1. n ≥ Consider a map h:Dom(h) C, where Dom(h) denotes domain of h. → Given a compact set K Dom(h) and an ε > 0, a neighborhood of h is ⊂ deﬁned as

(h,K,ε) := g :Dom(g) C K Dom(g), and sup g(z) h(z) <ε . N { → | ⊂ z K | − | } ∈ By a sequence h :Dom(h ) C (not necessarily defined on the same set) n n → converges to h we mean that given an arbitrary neighborhood of h defined as above, hn is contained in this neighborhood for sufficiently large n.

Inou-Shishikura class: Consider the cubic polynomial P (z)= z(1+z)2. This polynomial has a parabolic ﬁxed point at 0, a critical point at 1/3 which is − mapped to the critical value at 4/27, and another critical point at 1 which − − is mapped to 0. See Figure 4.1 75 Deﬁne

1 4 4π U := P − (B(0, e )) (( , 1] B) (4.1) 27 \ −∞ − ∪ 1 4 4π where B is the component of P − (B(0, e− )) containing 1. 27 −

Figure 4.1: A schematic presentation of the Polynomial P , its domain, and range. Similar colors and linestyles describe the map.

Deﬁne the class of maps

1 ϕ: U Uf is univalent,ϕ(0) = 0, ϕ′(0) = 1, and := P ϕ− : Uf C → . IS ◦ → φ 1 U¯ − extends onto f as a continuous function For a positive real number α , consider the following class ∗

[α ] := e2παi f f , and α [0,α ] . IS ∗ { | ∈ IS ∈ ∗ }

As the class [α ] is identiﬁed with the space of univalent maps on the unit IS ∗ disk with a neutral ﬁxed point at 0, it is a compact class in the above topology.

2παi Any map h = e f0 in [α ], with α = 0 and f0 , has a fixed point IS ∗ ∈ IS at 0 with multiplier e2παi, and another fixed point σ = 0. The σ fixed point h h has asymptotic expansion σ = 4παi/f ′′(0) + o(α), when h converges to f h − 0 0 in a fixed neighborhood of 0. Clearly, σ 0 as α 0. h → → See Figure 4.2 for the contents of the following theorem.

76 Theorem 4.3. (Inou-Shishikura [IS06]) There exist a real number α > 0, ∗ and positive integers k, kˆ such that the class [α ] satisﬁes the following: IS ∗

(1) h′′(0) =0 for any map h [α ]. ∈ IS ∗

(2) For any map h: Uh C in [α ], there exist a domain h Uh, → IS ∗ P ⊂ bounded by piecewise smooth curves, and a univalent map Φ : C h Ph → with the following properties:

(a) is compactly contained in U . Moreover, it contains the criti- Ph h cal point cp := ϕ( 1 ) in its interior as well as 0 and σ on its h − 3 h boundary.

(b) There exists a continuous branch of argument deﬁned on such Ph that

max arg(w) arg(w′) 2πkˆ w,w′ ∈Ph | − | ≤ (c) Φ ( ) = w C 0 Re(w) 1/α k . Also, Im Φ (z) h Ph { ∈ | ≤ ≤ ⌊ ⌋ − } h → + when z 0 and Im Φ (z) when z σ . ∞ ∈ Ph → h → −∞ ∈ Ph → h

(d) Φh satisﬁes the Abel functional equation, that is,

Φ (h(z))=Φ (z)+1, whenever z and h(z) belong to . h h Ph

Moreover, Φh is unique once normalized to send cph to 0.

(e) The map Φh depends continuously on h.

We refer to the univalent map Φh obtained in the above theorem as Per- turbed Fatou coordinate or sometimes Fatou Coordinate for short.

Remark. Parts (b) and (c) in the above theorem (existence of uniform k and kˆ) are not stated in [IS06] but it follows from their work. These are consequences of the compactness of the class [α ] and will become clear by Lemma 4.16 IS ∗ which we prove later. 77 Φh

0 1 cph b 1234 k α −

σh

Figure 4.2: An example of a perturbed Fatou coordinate and its domain.

2παi Renormalization: Consider a map h: U C in e with α α∗ (α∗ h → IS ≤ is obtained in Theorem 4.3) and let Φ : C be the normalized Fatou h Ph → coordinate obtained in that theorem. Deﬁne

:= z h :1/2 Re(Φh(z)) 3/2 , 2 < Im Φh(z) 2 , and C { ∈ P ≤ ≤ − ≤ } (4.2) ♯ := z :1/2 Re(Φ (z)) 3/2 , 2 Im Φ (z) . C { ∈ Ph ≤ h ≤ ≤ h } By deﬁnition, contains the critical value of h in its interior, and ♯ contains C C ♯ k 0 (ﬁxed point of h) on its boundary. For integers k > 0, let ( )− denote the C k ♯ unique connected component of h− ( ) with 0 on its boundary. Similarly, if C k there exists a unique connected component of h− ( ) which has non-empty C ♯ k k intersection with ( )− , it will be denoted by − . Let k be the smallest C C h k ♯ k positive integer (if it exists) for which the sets − h and ( )− h are contained C C in the set z 0 < Re Φ (z) < 1/α k 1/2 . { ∈ Ph | h − − }

78 We deﬁne the set Sh as,

k ♯ k S := − h ( )− h . h C ∪ C Consider the map

k 1 Φ h h Φ− : Φ (S ) C. (4.3) h ◦ ◦ h h h →

By equivariance property of Φh (Abel functional equation), this map projects − 4 2πiw 2π 1 i 2 via z = − e to a map of the form z e α z + O(z ), defined on some 27 → neighborhood of the origin. Further conjugating this map by s: z z¯, to make the rotation number → 2π i 2 at 0 positive, we obtain a map (h) of the form z e α z + O(z ). The map R → (h) is called the near parabolic renormalization of h by Inou and Shishikura. R We simply refer to it as renormalization of h. One can see (Lemma 4.5) that one time iterating (h) corresponds to several times iterating the map h, or R in other words, many times iterating h is equal to composition of two changes of coordinate and one iterate of (h). R The following theorem in [IS06] states that the above definition of near parabolic renormalization can be carried out for maps in . For a given R IS positive integer N, let IrrN denote the set of real numbers α = [a0, a1, a2,... ] with a N. i ≥ Theorem 4.4. (Inou-Shishikura) There exist an integer N > 0 such that if h e2παi with α Irr , then (h) is well-defined and belongs to the class ∈ IS ∈ N R 2π i 1 [1/N] (i.e. it can be written of the form (h)= e α P ψ− ). IS R ◦ The same conclusion holds for the map P (z)= e2παiz + z2, that is, (P ) α R α is well-defined and belongs to [1/N] provided α is small enough and belongs IS to IrrN .

2παi 2 Although quadratic polynomials Pα = e z +z do not belong to the class 79 [α ], the theorem states that one can deﬁne the Fatou coordinate for this IS ∗ map and renormalize it by the above deﬁnition. Hence, the theorem guaranties

that for α in IrrN , the sequence of renormalizations

f := n(P ): U C n R α n → are deﬁned and belong to the class [1/N]. For simplicity of notation, we IS let f0 = Pα and α0 = α. Each map fn has a ﬁxed point at 0 with multiplier e2παni.

4.1.3 Sectors around the post-critical set

Here we introduce a sequence of subsets of C containing 0 on their boundary, such that a.e. z in the Julia set of Pα has to visit these sets. From now on we will assume that N is large enough or α = 1/N is small enough so that the ∗ class [1/N] satisﬁes the conditions in Theorems 4.3 and 4.4. Moreover, for IS technical reasons, we will assume that 1 α (4.4) ∗ ≤ k + kˆ for k and kˆ obtained in Theorem 4.3.

Change of coordinates: For every n 0, let Φ = Φ denote the Fatou ≥ n fn coordinate of the map f : U C defined on the set := introduced in n n → Pn Pfn Theorem 4.3. By part (b) of Theorem 4.3 and our assumption (4.4), there are holomorphic inverse branches, η : C, of the map n Pn → 4 2πiz Exp(z) := z − s e : C C∗, where s(z)=¯z, −→ 27 ◦ → with ηn( n) Φn 1( n 1). There may be several choices for this map but we P ⊂ − P − choose one of them for each n and fix this choice whenever we refer to this map. 80 1 Now we can define ψn := Φn− 1 ηn : n n 1. Note that each ψn can − ◦ P → P − be continuously extended to 0, on the boundary of , by mapping it to 0. Pn Consider the maps

Ψ := ψ ψ ψ : n 1 ◦ 2 ◦◦ n Pn → P0 with values in the dynamical plane of the polynomial f0. For each f , n = 0, 1, 2,... , let and ♯ be the sets obtained in (4.2) n Cn Cn kn ♯ kn for f . Let k be the smallest positive integer for which − and ( )− are n n Cn Cn contained in . We deﬁne the sector S0 as Pn n

0 kn ♯ kn S := − ( )− . n Cn ∪ Cn ⊂ Pn

kn 0 By deﬁnition, the critical value of fn is contained in fn (Sn). For every n 0 deﬁne ≥

S1 := ψ (S0 ) . n n+1 n+1 ⊂ Pn

In general for i 2, let ≥

Si := ψ ψ (S0 ) . n n+1 ◦◦ n+i n+i ⊂ Pn

i i All these sectors contain 0 on their boundary. We will mainly consider S0, S1, 0 and Si for i =0, 1, 2,... . See Figure 4.3.

Lemma 4.5. Let z n 1 be a point with w:= Exp Φn 1(z) Dom(fn). ∈ P − ◦ − ∈ 1 There exists an integer ℓz with 2 ℓz k + kn 1 +1 such that ≤ ≤ ⌊ αn−1 ⌋ − −

ℓz fn 1(z) n 1, • − ∈ P −

ℓz Exp Φn 1(fn 1(z)) = fn(w), • ◦ − −

1 kn−1+ k 1 2 ℓz ⌊ αn−1 ⌋− − i 0 z, fn 1(z), fn 1(z),...,fn 1(z) i=0 fn 1(Sn 1), • − − − ∈ − − 81 ♯ 1 ( )− C

b cp b 0 cv Sn

1 − C

Figure 4.3: Figure shows the ﬁrst generation of sectors. The grey curve ap- proximates orbit of the critical point.

Moreover, if w int(Dom (f )), then ∈ n

2 ℓz z, fn 1(z), fn 1(z),...,fn 1(z) − − − 1 kn−1+ k 2 ⌊ αn−1 ⌋− − i 0 belong to the interior of i=0 fn 1(Sn 1) − − Proof. As w Dom(fn), by deﬁnition of renormalization (fn 1)= fn, there ∈ R − are

0 ζ Φn 1(Sn 1), ζ′ Φn 1( n 1) ∈ − − ∈ − P − such that

kn−1 1 Exp(ζ)= w, Exp(ζ′)= fn(w), ζ′ = Φn 1 fn 1 Φn− 1(ζ). − ◦ − ◦ −

0 Since Exp(Φn 1(z)) = w too, and ζ Φn 1(Sn 1), then there exists an − ∈ − − 1 integer l1 with kn 1 +1 l1 k such that Φn 1(z)+ l1 = ζ (k is − − ≤ ≤ ⌊ αn−1 ⌋ − − the constant obtained in Theorem 4.3). 82 By equivariance property of Φn 1, we have −

kn−1 1 ζ′ = Φn 1 fn 1 Φn− 1(ζ) − ◦ − ◦ − kn−1 1 = Φn 1 fn 1 Φn− 1(Φn 1(z)+ l1) − ◦ − ◦ − − kn−1+l1 1 = Φn 1 fn 1 Φn− 1(Φn 1(z)) − ◦ − ◦ − − kn−1+l1 = Φn 1 fn 1 (z). − ◦ −

If we let ℓz := kn 1 + l1 + 1 then we have −

1 ℓz 1 2 ℓz kn 1 + k +1, fn 1(z) = Φn− 1(ζ′ + 1) n 1, ≤ ≤ − ⌊αn 1 ⌋ − − − ∈ P − − and

ℓz 1 Exp Φn 1(fn 1(z)) = Exp Φn 1(Φn− 1(ζ′ +1)) = Exp(ζ′ +1) = fn(w) ◦ − − ◦ − −

This ﬁnishes the ﬁrst two properties. For the last property, we can see that one of the following two occurs

j 0 – there exists a positive integer j with fn 1(z) Sn 1, − ∈ −

j 0 – there exists a non-negative integer j with z fn 1(Sn 1). ∈ − −

If the ﬁrst case occurs (this is when l is positive), then

1 kn−1+ k 1 ⌊ αn−1 ⌋− − j 1 i 0 z, fn 1(z),...,fn−1(z) fn 1(Sn 1), − − ∈ − − i=kn−1 kn−1+1 j ℓz i 0 fn 1(z),...,fn 1(z) fn 1(Sn 1). − − ∈ − − i=0 If the second case occurs (when l is negative), then

kn−1+1 ℓz i 0 z, fn 1(z),...,fn 1(z) fn 1(Sn 1). − − ∈ − − i=j

83 The ﬁnal statement follows from open mapping property of holomorphic maps, that is, image of every open set under a holomorphic map is open. For

0 example, if w int(Dom (fn)) then ζ int(Φn 1(Sn 1)) which implies that z ∈ ∈ − − belongs to the interior of the above union.

In the above lemma there are clearly many choices for ℓz. Indeed, there

1 are k 1 choices for ℓz, however, in the following two lemmas we will ⌊ αn−1 ⌋ − − ℓz make a speciﬁc choice of fn 1(z) in order to control ℓz. −

Lemma 4.6. For every n 1, the two maps ≥

f : f ( ) and f qn : Ψ ( ) f qn (Ψ ( )) n Pn → n Pn 0 n Pn → 0 n Pn are conjugate by Ψn, that is, the following diagram

qn f0 qn Ψn( n) / f0 (Ψn( n)) OP O P Ψn Ψn

fn / f ( ) Pn n Pn commutes wherever it is defined. Similarly for every n > m, f : f ( ) is conjugate to some iterate n Pn → n Pn of f defined on the set ψ ψ ( ). m m+1 ◦◦ n Pn Proof. By definition of renormalization , this property holds near 0 (fixed R point). so, by analytic continuation, they hold on their domain of definition.

The integers qn are the closest return times for the rotation of angle α0 near the 0 ﬁxed point.

Consider the map f kn : S0 f kn (S0) . n n → n n ⊆ Pn The following lemma translates dynamics of this map to the dynamic plane of

f0. 84 Lemma 4.7. The map f kn : S0 f kn (S0) is conjugate to n n → n n

f knqn+qn−1 : Ψ (S0) Ψ (f kn (S0)), 0 n n → n n n

by Ψn.

Proof. Proof of this lemma is similar to the previous one. In this case knqn +

0 qn 1 is the return time for Ψn(Sn) back to Ψn( n) under f0. − P On each level j 0, we consider union of sectors ≥ 1 kj + k 1 ⌊ αj ⌋− − 0 i j Ωj := fj (S0). i=0 Using the two previous lemmas, we transfer these sets to the dynamic plane of

f0 to obtain, qn+1+(kn k 1)qn − − n i n Ω0 := f0(S0 ), i=0 for every n 0. ≥ 0 To transfer the sectors in Ωn from level n to level 0, the ﬁrst kn sectors 1 give knqn + qn 1 sectors by Lemma 4.7, and the k 1 remaining ones − ⌊ αn ⌋ − − 1 produce qn( k 1) by Lemma 4.6. Thus, totally we obtain ⌊ αn ⌋ − − 1 1 (knqn + qn 1)+ qn( k 1) = qn( + qn 1)+ qn(kn k 1) − ⌊αn ⌋ − − ⌊αj ⌋ − − − q + q (k k 1), ≤ n+1 n n − −

by formula qn+1 = an+1qn + qn 1. −

Proposition 4.8. For every n 0, we have the following: ≥

n+1 n 1. Ω0 is compactly contained in the interior of Ω0

n 2. The post critical set of f0 is contained in the interior of Ω0

85 Proof. Part (1) : To show that Ωn+1 Ωn, it is enough to show that for every 0 ⊂ 0 z Sn+1 there are points z , z ,...,z in Sn as well as non-negative integers ∈ 0 1 2 m 0 1 t1, t2,...,tm+1, for some positive integer m (indeed m = kn+1 + k 1), ⌊ αn+1 ⌋− − with the following properties:

t1 tm+1 qn+2+(kn+1 k 1)qn+1 f (z )= z and f (z )= f − − (z), • 0 1 0 m 0

tj f0 (zj 1)= zj, for j =2, 3,...,tm 1, • − −

t q +(k k 1)q , for every j =1, 2,...,m + 1. • j ≤ n+1 n − − n

n+1 1 0 0 For a given z S , let ζ := Ψ− (z) S . By deﬁnition of S , the ∈ 0 n+1 ∈ n+1 n+1 iterates 1 kn+1+ k 1 2 ⌊ αn+1 ⌋− − ζ, fn+1(ζ), fn+1(ζ),...,fn+1 (ζ)

are deﬁned and belong to the domain of fn+1.

By Lemma 4.5 for ψn+1(ζ), there are two points ξ1 (take ξ1 = ψn+1(ζ)) and ξ in as well as a positive integer ℓ with Exp Φ (ξ ) = ζ, Exp 2 Pn 0 ◦ n 1 ◦ Φ (ξ ) = f (ζ) and f ℓ0 (ξ ) = ξ . Let σ S0 and an integer ℓ with n 2 n+1 n 1 2 1 ∈ n 1 1 ℓ k + a k 1 be such that f ℓ1 (σ )= ξ . With the same lemma, ≤ 1 ≤ n n+1 − − n 1 1 there is a point σ2 in the orbit

2 ℓ0 1 ξ1, fn(ξ1), fn(ξ1),...,fn − (ξ1),ξ2

0 1 which belongs to S . Let ℓ2 be the positive integer with 1 ℓ2 kn + n ≤ ≤ ⌊ αn ⌋ − k 1 that f ℓ2 (σ )= σ . − n 1 2 By the same argument for ξ with Exp Φ (ξ )= f (ζ), we obtain points 2 ◦ n 2 n+1 σ S0, ξ and a positive integer ℓ with 1 ℓ k + a k 1, 3 ∈ n 3 ∈ Pn 3 ≤ 3 ≤ n n+1 − − ℓ3 such that fn (σ2)= σ3.

86 1 Repeating this argument for ξ3,ξ4,...,ξm, for m = kn+1 + k 1, ⌊ αn+1 ⌋ − − one obtains a sequence

σ1, σ2,...,σm

0 of points in Sn and positive integers

1 ℓ1,ℓ2,...,ℓkn+1+ k 1,ℓm+1, ⌊ αn+1 ⌋− −

1 all bounded by kn + k 1, which satisfy: ⌊ αn ⌋ − −

ℓj+1 fn (σ )= σ for all j =2, 3,...,m 1 • j j+1 −

f ℓ1 (σ )= ξ and f ℓm+1 (σ )= ξ • n 1 1 n m m

Now we deﬁne z := Ψ (σ ) Sn, for j = 1, 2,...,m. By deﬁnition i n i ∈ 0 qn+2+(kn+1 k 1)qn+1 Ψn(ξ1)= z. We claim that Ψn(ξm)= f0 − − (z). This is because

1 kn+1+ k 1 Exp Φ (ξ )= f ⌊ αn+1 ⌋− − (ζ) ◦ n m n+1

qn+2+(kn+1 k 1)qn+1 which is mapped to f0 − − (z) byΨn+1 using Lemmas 4.6 and 4.7

for f0 and fn+1.

By Lemmas 4.6 and 4.7, ℓj times iterating fn corresponds to tj times

iterating f0, for each j = 1, 2,...,ℓm+1. With the same lemmas, as ℓj is

1 bounded by kn + k 1, each tj is bounded by ⌊ αn ⌋ − − 1 knqn + qn 1 +( k 1)qn qn+1 + qn(kn k 1). − ⌊αn ⌋ − − ≤ − −

n+1 n To show that Ω0 is compactly contained in the interior of Ω0 , we use the open mapping property of holomorphic maps. So if z′ is a point in the closure

n+1 of Ω0 , as f0 is a polynomial deﬁned on the whole complex plane, there exists

n+1 t0 a point z in the closure of S0 with f0 (z) = z′ , for a non-negative integer t less than q +(k k 1)q . The last statement in Lemma 4.5 implies 0 n+2 n − − n+1 87 that all the points σj in the above argument can be chosen in the interior of

0 n Sn. Hence, all the zi are contained in the interior of S0 .

Part (2) : First we claim that for every n 0, the critical point of f ≥ 0 belongs to Ωn and can be iterated at least (a k 1)q times with in this 0 n+1 − − n set.

To prove the claim, note that f : S0 f kn (S0) is a degree two map. Thus, n n → n n by Lemma 4.7, f knqn+qn+1 : Sn Ψ (f kn (S0)) is also a degree two map. That 0 0 → n n n means that the critical point of f is contained in the union knqn+qn−1 f i(Sn). 0 ∪i=0 0 0 n Therefore, by deﬁnition of Ω0 , the critical point can be iterated at least

qn+1 +(kn k 1)qn knqn qn 1 =(an+1 k 1)qn − − − − − − −

n times with in Ω0 . As a k 1 1 and q growth (exponentially) to infinity as n goes to n+1 − − ≥ n infinity, combining with part (1), the critical point of f0 can be iterated infinite

n number of times with in each Ω0 . For every n 0, Ωn contains closure of Ωn+1 in its interior and Ωn+1 ≥ 0 0 0 contains orbit of the critical point. Therefore, the post-critical set is contained

n in Ω0 .

n In the next lemma we show that all the sectors contained in the union Ω0

are visited by almost every point in the Julia set of f0, that is,

Lemma 4.9. Let n and ℓ be positive integers with 0 ℓ q +(k k 1)q . ≤ ≤ n+1 n − − n Then for almost all z in the Julia set of f0, there exists a non-negative integer ℓ such that f ℓz (z) f ℓ(Sn). z 0 ∈ 0 0

Proof. Obviously, it is enough to prove the lemma for ℓ = 0. We claim that for every n 0, the set of points which visit Ωn+1 contains the set of points ≥ 0

88 n which visit S0 . Assuming the claim for a moment, because the set of points in n+1 the Julia set that visit Ω0 has full measure by Propositions 4.2 and 4.8, we can conclude the lemma. To prove the claim, let z be an arbitrary point in J for which there exists an integer t 0 with f t1 (z) Ωn+1. Let t t be a positive integer with 1 ≥ 0 ∈ 0 2 ≥ 1

t2 qn+2+(kn+1 k 1)qn+1 n+1 f (z) f − − (S ). 0 ∈ 0 0

1 t2 1 t2 Let ξ denote the point Ψ− (f (z)) in . As ζ := Ψ− (f (z)) = Exp Φ (ξ) n 0 Pn n+1 0 ◦ n belongs to , f (ζ) is deﬁned. Hence, by Lemma 4.5, there exists a non- Pn+1 n+1 negative integer j with the points ξ, f (ξ), f 2(ξ),...,f j(ξ) contained in and n n n Pn j 0 t2+qnj the last point, fn(ξ), contained in Sn. By Lemma 4.6, f0 (z) belongs to n S0 .

4.1.4 Size of the sectors

n Now we want to control size of certain sectors contained in the unions Ω0 in terms of Brjuno function. The following two lemmas are our main technical tools. Their proof comes at the end of this section.

Lemma 4.10. There exists a constant M 1 such that for every integer n 1 ≥ ≥ there exists an integer τ(n) with k τ(n) a k 2, and n ≤ ≤ n+1 − −

diam(f τ(n)(S0)) Mα . n n ≤ n

Lemma 4.11. There exists a constant M 1 such that for every integer ≥ n 1, there exists an integer κ(n), with 0 κ(n) a k 1 that satisﬁes ≥ ≤ ≤ n+1 − − the following: For every w , ∈ Pn+1

κ(n) (1) fn ψ (w) , ◦ n+1 ∈ Pn 89 κ(n) αn (2) fn ψ (w) Mα w . | ◦ n+1 | ≤ n| |

From now on we assume that M denotes a constant which satisﬁes these two lemmas.

Proposition 4.12. There exists a constant C such that for every m 1, there ≥ exist non-negative integers γ(m) and γ′(m) q +(k k 1)q for which ≤ n+1 n − − n the following holds

(1) diam(f γ(m)(Sm)) C α αα1 αα1α2 αα1α2α3 ...αα1...αm−1 . 1 1 ≤ 1 2 3 4 m

(2) f γ(m)(Sm) 1 1 ⊆ P1

′ γ(m) m γ (m) m+1 γ(m) m (3) ψ1(f1 (S1 )) = f0 (S0 ), that is, ψ1(f1 (S1 )) is among the sec- m+1 tors in the union Ω0 .

Proof. For the constant M obtained for the two Lemmas 4.10 and 4.11, let

C = M M α1 M α1α2 M α1α2α3 ... = M 1+α1+α1α2+α1α2α3+...

2 3 M 1+1/2+1/2 +1/2 +... = M 2 < . (as α < 1/2) ≤ ∞ i

Given m 1, by Lemma 4.10, there exists τ(m) with k τ(m) ≥ m ≤ ≤ k k 2, and m − − diam(f τ(m)(S0 )) M α . m m ≤ m τ(m) 0 By Lemma 4.11 with n = m 1, and w fm (S ), we obtain − ∈ m

κ(m 1) τ(m) 0 αm−1 diam(fm 1− ψm(fm (Sm))) M αm 1 (M αm) . − ◦ ≤ −

Now by Lemma 4.6 we have

κ(m 1) τ(m)am+1 +1 0 αm−1 αm−1 diam(fm 1− fm 1 (ψm(Sm)) M M αm 1 αm − ◦ − ≤ − 90 or equivalently

κ(m 1)+τ(m)am+1 +1 1 αm−1 αm−1 diam(fm 1− (Sm 1) M M αm 1 αm . − − ≤ −

Again applying Lemma 4.11 with n = m 2, the last inequality implies − that

κ(m 2) κ(m 1)+τ(m)am+1 +1 1 diam(fm 2− ψm 1(fm 1− (Sm 1)) − ◦ − − − ≤ αm−1 αm−1 αm−2 M αm 2 (M M αm 1 αm ) . − − which is equivalent, by Lemma 4.6, to

κ(m 2)+(κ(m 1)+τ(m)am+1 +1)am+1 1 diam(fm 2− − (ψm 1(Sm 1))) − − − ≤ αm−1 αm−1 αm−2 M αm 2 (M M αm 1 αm ) . − −

Repeatedly using Lemma 4.11 with n = m 3, m 4,..., 1, one obtains − −

diam(f γ(m)(Sm)) M α [M α [M α [... [M α ]αm−1 ]αm−2 ... ]α2 ]α1 1 1 ≤ 1 2 3 m

γ(m) m for some integer γ(m). Therefore f1 (S1 ) has diameter less than

M M α1 M α1α2 ...M α1α2...αm−1 α αα1 αα1α2 αα1α2α3 ...αα1...αm−1 1 2 3 4 m ≤ C α αα1 αα1α2 αα1α2α3 ...αα1...αm−1 . 1 2 3 4 m this finishes the first Part of the proposition. The second part of the proposition follows from above argument when Lemma 4.11 was used with m = 1. To see the third statement in the proposition, first note that τ(m) is chosen strictly less than a + k k 1. Therefor, ψ (f γ(m)(Sm)) is among the m+1 m − − 1 1 1 m+1 sectors in the union Ω0 . Indeed, one can see that γ′(m) is strictly between kmqm + qm 1 and qm+1 +(km k 1)qm. − − − 91 Lemma 4.13. The sequence

α αα1 αα1α2 αα1α2α3 ...αα1...αk−1 { 1 2 3 4 k }k

converges to zero as k , if and only if the Brjuno sum →∞

∞ log qn+1 q n=0 n is divergent.

This is a purely combinatorial lemma and one may refer to [Yoc95] (page 13) for its proof.

Proof of Theorem 1.3. The set of points in the Julia set of f0 which accumulate on the 0 ﬁxed point is equal to the intersection of the sets

A = z J : (z) B(0, 1/n) = ∅ . n { ∈ O ∩ } for n = 1, 2,... . To prove the theorem, it is enough to show that every An has full Lebesgue measure in the Julia set. As the map ψ : has 1 P1 → P0 continuous extension to the boundary point 0, there exists a δn > 0 such that if w < δ , for some w , then ψ (w) < 1/n. | | n ∈ P1 | 1 | By Lemma 4.13, there exists an integer m> 0, for which

C α αα1 αα1α2 αα1α2α3 ...αα1...αk−1 1 2 3 4 k

is less than δn, where C is the constant obtained in Proposition 4.12. Now by

γ(m) m part (1) and (2) of Proposition 4.12, ψ1(f1 (S1 )) is contained in B(0, 1/n). Part (3) of Proposition 4.12 and Lemma 4.9 implies that this set is visited

by almost every point in the Julia set of f0. This completes our proof of Theorem 1.3.

92 4.1.5 Perturbed Fatou coordinate

In order to prove Lemmas 4.10 and 4.11 in this section, we will give an

approximate formula for the Fatou coordinate Φh with a bound on its error.

2παi 1 2παi Assume h(z)= e P ϕ− (z): ϕ(U) C belongs to the class e , ◦ → IS and σh denotes its non-zero ﬁxed point. In [IS06], N was chosen large enough

so that h(z) has only two ﬁxed points 0 and σh in its domain of deﬁnition. Therefore, one can write h(z) as

h(z)= z + z(z σ )u (z) − h h

where uh(z) is a non-zero holomorphic function deﬁned on the set ϕ(U). Dif- ferentiating both sides of this equation at 0, one obtains

1 e2παi σh = − . (4.5) uh(0)

Note also that the map u (z)=(h(z) z)/(z(z σ )) depends continuously h − − h on the map h(z). Let σ τ (w) := h h 1 e 2πiαw − − be the universal covering of the Riemann sphere minus two points 0 and σh that has deck transformation group generated by

1 T (w) := w + . α α

One can see that τ (w) 0, as Im(αw) , and τ (w) σ , as Im(αw) h → →∞ h → h → . −∞ Deﬁne the map

1 σhuh(z) Fh(w) := w + log 1 , with z = τα(w) 2παi − 1+ zuh(z)) 93 on the set of points w with τ (w) Dom(h). The branch of log in the above h ∈ formula is determined by π < Im log( ) < π. The map F is deﬁned on the − h inverse image of Dom (h) under τα. It is immediate calculation to see that

h τ = τ F , and T F = F T . ◦ h h ◦ h α ◦ h h ◦ α

Indeed, Fh was deﬁned using this relations. We will see in a moment that the Fatou coordinate of a map in the class

[α ] is “essentially” equal to τα for our purposes. Hence, we wish to further IS ∗ control τh on certain domains. For every real number R> 0, let Θ(R) denote the set

Θ(R) := C T n(B(0, R)). \ α n Z ∈

Lemma 4.14. There exists a positive constant C1 such that

(1) For every Y > 0, there exists ε > 0, such that for every h e2παi , Y ∈ IS with α<εY , we have

w Θ(Y ), τ (w) C /Y ∀ ∈ | h | ≤ 1

(2) For every r (0, 1/2), w Θ( r ), and every h [α ] we have ∈ ∈ α ∈ IS ∗

α 2πα Im w τ (w) C e− . | h | ≤ 1 r

Proof. The σh ﬁxed point has the form (4.5) in terms of uh, where uh is a non-zero function in a compact class. Therefore, there exists a constant C′

2παi such that for every h e , σ < C′α. The rest follows from analyzing ∈ IS | h| 2πiαw the explicit formula 1/(1 e− ) on Θ(Y ). −

94 To see Part (1), ﬁx an arbitrary Y > 0. If w Θ(Y ), then 1 e2πiαw ∈ − 2παY belongs to complement of the ball of radius e− 1 centered at 0 in C. − Therefore σ C α C h < ′ = ′ |1 e 2παY | 3παY 3πY − − for small α.

2πiαw 2πr To see part (2), one can observe that for such a w, 1 e− e 1, | − | ≥ − and conclude that there exists a constant C′′ with

2πiαw 2πα Im w 1 e− C′′re . | − | ≥

This implies that

C′ α 2πα Im w τh(w) e− . | | ≤ C′′ r C′ C′ Now we can take C1 as the maximum of 3π and C′′ .

Lemma 4.15. There exist positive constants ε0 > 0, C2, C3, C4 as well as a positive integer j , such that for every map h e2παi , with α ε , the 0 ∈ IS ≤ 0 induced map Fh is deﬁned and is univalent on Θ(C2), and moreover

(1) For all w Θ(C ), we have ∈ 2

F (w) (w + 1) < 1/4, and F ′ (w) 1 < 1/4. | h − | | h − |

(2) For every r (0, 1/2), and w Θ( r + 1), we have ∈ ∈ α

α 2πα Im w α 2πα Im w F (w) (w +1)

(3) cp B(0, 2) B(0,.22). If i(h) is the smallest non-negative integer with h ∈ \ Re F i(h)(cp ) C , then i(h) min j , 1/α . h Fh ≥ 2 ≤ { 0 } (4) For every positive integer j j + 2 , we have ≤ 0 3α

Im F j(cp ) C (1 + log j), and Re F j(cp ) j C (1 + log j). | h Fh | ≤ 4 | h Fh − | ≤ 4 95 Proof.

2παi 1 Parts (1) and (2) : Consider a map h = e P ϕ− : ϕ(U) C in ◦ → e2παi . As cp = 1/3 / B(0, 1/3), by Koebe Distortion Theorem, cp = IS P − ∈ h φ( 1/3) / B(0, 1/12). So, every h in the above class is deﬁned and univalent − ∈ on B(0, 1/12). Applying part (1) of Lemma 4.14 with Y = 12C1, we obtain an ε > 0 such that if α ε , then 0 ≤ 0

τ (Θ(Y )) B(0, 1/12). h ⊂

Therefore, the induced map Fh is deﬁned and univalent on Θ(Y ). For w Θ(Y ), using notation λ = e2παi, we have ∈

1 σhuh(z) Fh(w)= w + log 1 , with z = τh(w) 2παi − 1+ zuh(z) 1 1 σ u (z) = w +1+ log (1 h h ) . 2παi λ − 1+ zuh(z) Now assume we want to show that

1 1 σhuh(z) Fh(w) (w + 1) = log (1 ) < A | − | 2πα λ − 1+ zuh(z) for some A with 0

1 1 σ u (z) A (1 h h ) 1 < . 2πα λ − 1+ zuh(z) − e As λ = 1, it is enough to show that | | 1 σ u (z) A 1 h h λ < . 2πα − 1+ zuh(z) − e Replacing σ by its value from equation (4.5), and using 1 λ < 2πα, we h | − | obtain

1 u (z) u (z) (1 λ)(1 h ) < 1 h . 2πα − − (1 + zuh(z))uh(0) − (1 + zuh(z))uh(0) 96 Since uh belongs to a compact class, it is possible to make

u (z) 1 h (4.6) − (1 + zuh(z))uh(0)

1 less than 4e by restricting z to a sufficiently small disk of radius δ around 0. So, u (z) 1 1/4 1 h < = , on B(0, δ). − (1 + zuh(z))uh(0) 4e e By Lemma 4.14, part (1), there is a constant C′(δ) Y such that z = ≥ | | τ (w) < δ holds for every w Θ(C′(δ)). With this constant C′(δ) (for C ), | α | ∈ 2 we have the first inequality in part (1) of the lemma. The second inequality in part (1) follows from the Cauchy estimate (integral formula) applied to F (w) w 1, once we restrict w to smaller domain h − − Θ(C′(δ) + 1). Hence, for C2 := C′(δ) + 1 we have both inequalities. For the first inequality in Part (2), using Taylor’s Theorem for Expres- sion (4.6), one obtains

u (z) u (0) 1 h < 2 h′ z . − (1 + zu (z))u (0) |u (0) 1|| | h h h −

Moreover, as uh belongs to a compact class,

uh′ (0) < D′ |u (0) 1| h −

for some constant D′. Using part (2) of Lemma 4.14, we conclude that for every w Θ( r ), we ∈ α have

uh(z) α 2πα Im w 1 < 2D′C1 e− | − (1 + zuh(z))uh(0)| r ′ 2D C1 α 2πα Im w e− = e r . e

This proves the ﬁrst inequality by introducing C3 := 2D′C1/e. 97 The other inequality in (2) is also a consequence of Cauchy estimate, once

r we restrict w to Θ( α + 1). 2πα Part (3) : By explicit calculation one can see that e− h is univalent on

2π the ball B(0, 1 8/27e− ) B(0, 2/3). Koebe distortion Theorem applied − ⊃ to this map on B(0, 2/3) implies that cp B(0, 2) B(0,.22). h ∈ \ 1 By above argument, there is a choice of cpFh in τα− (cph) that belongs to a compact subset of C (independent of α). Since h converges to maps in the compact class as α 0, cp visits Θ(C ) in a ﬁnite number of iterates IS → Fh 2 i(h), uniformly bounded by some constant j0 independent of h. For the same reason,

i F (cp ) C′, for i =0, 1,...i(h) | h Fh | ≤ for some constant C′. Part (4) : It is enough to prove the inequalities for small values of α. For larger α, there are only ﬁnite number of iterates to consider. Therefore, by our previous argument in part (3), the inequalities hold for large enough C4. So, in the following we assume that α min C2 + 5j0+5 , 1 . ≤ { 3 24 8C2+41 } By the ﬁrst part of this lemma, at each step j with i(h) j j + 2 , we ≤ ≤ 0 3α have F j(cp ) Θ(C ), h Fh ∈ 2

3j j 5 5j C′ + Re F (cp ) C′ + + . 4 ≤ h Fh ≤ 4 4

and,

1 j j j 1 C′ C′ Im F (cp ) C′ + C′ + . − − 6α ≤ − − 4 ≤ h Fh ≤ 4 ≤ 6α

jα j 2 Now, one can use part (2) with rj = 6 at Fh (cpFh ), for j = i(h),...,j0+ 3α ,

98 to obtain:

j j 6 2πα(C′+ 1 ) F (F (cp )) F (cp ) 1 C e 6α | h h Fh − h Fh − | ≤ 3 j ′ 1 1 6C eπ(C + 3 ) ≤ 3 j Putting above inequalities together using triangle inequality, we obtain the following estimates for every j j + 2 : ≤ 0 3α j j π(C′+ 1 ) 1 Im F (cp ) C′ +6C e 3 | h Fh | ≤ 3 m m=i(h) π(C′+ 1 ) C′ +6C e 3 (1 + log j) ≤ 3 π(C′+ 1 ) 2 C′ +6C e 3 (1 + log ), ≤ 3 3α similarly,

j π(C′+ 1 ) Re F (cp ) C′ +(j i(h))+6C e 3 (1 + log j), and h Fh ≤ − 3 j π(C′+ 1 ) Re F (cp ) C′ +(j i(h)) 6C e 3 (1 + log j j ) h Fh ≥ − − − 3 | − 0|

This ﬁnishes part (4) by introducing the appropriate constant C4.

The following lemma is basically repeating existence of Fatou coordinate in the second part of Theorem 4.3. There is a standard argument based on the measurable Riemann mapping theorem to construct such a coordinate. Since we need to further analyze this coordinate, we repeat this argument in the next lemma.

For real constant Q> 0, let ΣQ denote the set

1 Σ := w C : Q Re(w) Q Q { ∈ ≤ ≤ α − }∪ w C : Re(w) Q, and Im w Re(w)+2Q { ∈ ≤ | | ≥ − }∪ 1 1 w C : Re(w) Q, and Im w Re(w) +2Q . { ∈ ≥ α − | | ≥ − α }

99 Im z ΣC2 la Fh

b b

O Re z

Figure 4.4: The gray region shows the domain ΣC2 .

Lemma 4.16. For every map h e2παi with α less than ε (obtained in ∈ IS 0 Lemma 4.15), there is a univalent map L : Dom(L ) C with the following h h → properties:

(1) Σ cp Dom(L ) and C2 ∪{ Fh } ⊂ h

w C :0 Re(w) 1/α k L (Dom(L )) { ∈ ≤ ≤ ⌊ ⌋ − } ⊆ h h

(same k as in Theorem 4.3).

(2) Lh satisﬁes

Lh(Fh(w)) = Lh(w) + 1 (Abel functional equation)

whenever both sides are deﬁned. Moreover, Lh is unique once normalized

by mapping the critical value of Fh to 1.

Proof. Let l denote the vertical line a + it :

g : w C :0 Re(w) 1 { ∈ ≤ ≤ }→Kh deﬁned as g(s + it) := (1 s)(a + it)+ sF (a + it). − h The partial derivatives of g exist everywhere and can be calculated as

∂g 1 ∂g ∂g (s + it)= [ i ](s + it) ∂w 2 ∂s − ∂t 1 = [Fh(a + it) (a + it)+1+ s(Fh′ (a + it) 1)], 2 − − (4.7) ∂g 1 ∂g ∂g (s + it)= [ + i ](s + it) ∂w¯ 2 ∂s ∂t 1 = [F (a + it) (a + it) 1+ s(1 F ′ (a + it))]. 2 h − − − h By the inequalities in part (1) of Lemma 4.15, dilatation of the map g, g /g , is bounded by 1/3. Thus, it is a quasi-conformal map onto . More- | w¯ w| Kh over,

1 1 w l , g− (F (w)) = g− (w)+1. ∀ ∈ a h The Beltrami diﬀerential

∂g/∂w¯ dw¯ ν(w) := (w) ∂g/∂w dw

is the pull back of the standard complex structure on C by g. Using ν(T1(w)) = ν(w), we can extend ν(w) over the whole complex plane C. By measurable Riemann mapping theorem ([Ah66], Ch V, Theorem 3), there exists a unique quasi-conformal mapping g : C C which solves the Beltrami diﬀerential 1 → equation ∂ g = ν ∂ g and leaves the points 0 and 1 ﬁxed. ∂w¯ 1 ∂w 1 101 1 1 As g T g− is quasi-conformal and ∂(g T g− )/∂w¯ = 0, by explicit 1 ◦ 1 ◦ 1 1 ◦ 1 ◦ 1 calculation, Weyl’s Lemma ([Ah66], Ch II, Corollary 2) implies that this map is

a conformal map of the complex plane. As it is conjugate to T1, it can not have

any ﬁxed point. Therefore, it is a translation of the plane. Finally, g1(0) = 1

1 implies that g T g− = T , or in other words, g (w +1)= g (w) + 1. 1 ◦ 1 ◦ 1 1 1 1 1 For the same reason, the map L := g g− : C is conformal and by h 1 ◦ Kh → previous arguments satisfies Lh(Fh(w)) = Lh(w)+1 on la. This relation can be used to extend Lh on a larger domain. By part (1) of Lemma 4.15, for every w Σ there is an integer j for which F jw (w) . Thus domain of L ∈ C2 w h ∈ Kh h contains at least ΣC2 and by definition satisfies the Abel functional equation on its domain of definition. Note that L (a) = 0. Given any simply connected domain in C 0, σ , h \{ h} there is a continuous inverse branch of τh defined on this domain. Further, if

1 image of such a domain under this branch, τh− , is contained in domain of Lh, composition of this branch and Lh is a Fatou coordinate for h. By uniqueness

1 in Theorem 4.3, there exists a constant t such that L τ − + t = Φ is the h h ◦ α h h unique Fatou coordinate which sends cph to zero. This would imply that Lh extends over a larger domain containing cpFh on its boundary. Moreover, its image contains the set

w C :0 Re(w) 1/α k { ∈ ≤ ≤ ⌊ ⌋ − } for the constant k obtained in that theorem.

1 To control Fatou coordinate of a given map h, which is of the from L τ − , h ◦ h we need to control Lh. First we give a rough estimate on derivative of Lh.

Lemma 4.17. There exists a positive constant C such that for every h 5 ∈ 2παi 1 e , and every ζ with 1 Re ζ 1/α k, we have 1/C (L− )′(ζ) IS ≤ ≤ − 5 ≤| h | ≤ C5. 102 1 Proof. Let G : (0, 1/α k) ( , ) C, denote the map L− through this − × −∞ ∞ → h proof. We will consider two separate cases. First assume ξ := G(ζ) Θ(C ) ∈ 2 and Im ζ (1.5, 1/α k 1.5). So, G is deﬁned and univalent on B(ζ, 1.5) ∈ − − and F (ξ) B(ξ +1, 1/4). Now, by 1/4 Theorem, G′(ζ) /4 (1+1/4) which h ∈ | | ≤ implies G′(ζ) 5. For the other direction, by Koebe distortion theorem, we ≤ have

w B(ζ, 1), G′(w)/G′(ζ) 45. ∀ ∈ | | ≤

By comparing distances d(ζ,ζ + 1) and d(ξ, Fh(ξ)), we obtain

45 G′(ζ) 1 sup G′(w) 1 1/4=3/4, | | ≥ w B(ζ,1) | | ≥ − ∈

which implies, G′(ζ) 1/36. This proves the lemma in this case. | | ≥ Now if ξ Θ(C ) and Im L (ξ) (1, 1/α k). By Abel functional equation ∈ 2 h ∈ − 2 1 2 in Lemma 4.16, at least one of ξ, Fh(ξ), Fh (ξ), Fh− (ξ), Fh− (ξ) satisﬁes above condition. Diﬀerentiating Abel functional equation and using Lemma 4.15, part (1), we see

3/4 L′ (F (ξ)) / L′ (ξ) = F ′ (ξ) 5/4, ≤| h h | | h | | h | ≤

which takes care of this case. Finally, if ξ / Θ(C ) then ξ belong to a compact subset of C. As the ∈ 2 normalized Fatou coordinate Lh is univalent and depends continuously on h in the compact open topology, this case follows from compactness of the class [α ]. IS ∗

Finally, the following is our ﬁne control of Lh. Let C6 > 1 be a positive constant that satisﬁes C (1 + log 5 )+2C C /α. 4 4α 5 ≤ 6

103 Lemma 4.18. There exists a positive constant C7 such that for every map Lh with α(h) <ε , every r (0, 1/2), and every w ,w Dom L with 0 ∈ 1 2 ∈ h – Re w = Re w , and Im w , Im w > C /α, 1 2 1 2 − 6

– for all t (0, 1), tw + (1 t)w Θ( r + 1), ∈ 1 − 2 ∈ α we have,

(1) Re(L (w ) L (w )) C /r | h 1 − h 2 | ≤ 7

(2) Im (L (w ) L (w )) Im(w w ) C /r | h 1 − h 2 − 1 − 2 | ≤ 7

Proof. Given w1,w2 satisfying the conditions in the lemma, choose a vertical line l with w ,w in the construction of the map g in Lemma 4.16. Let 1 2 ∈ Kh i(h) i(h)+j Fh (cpFh ) be the ﬁrst visit of cpFh to ΣC2 , and let w∗ := Fh (cpFh ) be the ﬁrst visit of this point to . By part (1) of Lemma 4.15, j 1/2α 2 . Kh ≤ 3/4 ≤ 3α Hence, (4) of the same lemma implies that

2 2 Im w∗ [ C (1 + log ),C (1 + log )]. (4.8) ∈ − 4 3α 4 3α

Let th be a complex constant with imaginary part in this set and

(Lh + th)(w∗)= i(h)+ j.

Then by Abel functional equation we conclude that cpFh is mapped to 0 under

Lh + th. We will denote this map by the same notation Lh, thus Lh(w∗) = i(h)+ j.

1 We have the following simple inequalities for the quasi-conformal map g− constructed using the choice of vertical line l:

1 1 Im (g− (w ) g− (w )) Im (w w ) 1/2, | 1 − 2 − 1 − 2 | ≤ 1 1 Re(g− (w )) Re(g− (w )) 1/2. | 1 − 2 | ≤ 104 1 To prove similar results for Lh, we will compare it to g− using Green’s

integral formula. Choose t1 and t2 so that w1 and w2 are contained in the curves s g(s + it ), and s g(s + it ), for 0 s 1, respectively. Using → 1 → 2 ≤ ≤ notations ζ = s + it, dζ = ds + idt and dζ¯ = ds idt, by Green’s Theorem − applied to the map g (ζ)= L g on the rectangle 1 h ◦

:= ζ C :0 Re(ζ) 1, t Im ζ t D { ∈ ≤ ≤ 1 ≤ ≤ 2}

we have

∂g1(ζ) ¯ g1(ζ) dζ = ¯ dζ dζ. (Green’s formula) ∂ − ∂ζ ∧ D D

If we let w = g(ζ), the complex chain rule for g1(ζ), using the Cauchy-

∂Lh Riemann equation ∂w¯ = 0, can be written as ∂g ∂(L g) ∂L ∂g 1 = h ◦ =( h g) . ∂ζ¯ ∂ζ¯ ∂w ◦ ∂ζ¯

Therefore,

t2 1 ∂g1(ζ) ¯ ∂g1(ζ) ¯ dζ dζ 2 ¯ dsdt ∂ζ ∧ ≤ t1 0 ∂ζ D t2 1 4 α 2πα Im g(s+it) sup L′ C e− dsdt. ≤ 3 | h| 3 r t1 0 The last inequality follows from (4.7) and Lemma 4.15–(2). By our assumption

on w1 and w2, the last integral is less than or equal to

∞ 4 α 2πα(t 1/4) C5C3 e− − dt C6/α 3 r − 4C5C3 2πα( C6/α 1/4) e− − − ≤ 3πr 4C C eπ(2C6+1) 1 5 3 , ≤ 3π r

which is bounded independent of α.

105 If we parametrize boundary of as D ϑ (ℓ) := iℓ, ℓ [t , t ] ϑ (ℓ) := ℓ + it , ℓ [0, 1] 1 ∈ 1 2 2 2 ∈ ϑ (ℓ) := 1 + i(t + t ℓ), ℓ [t , t ] ϑ (ℓ) := 1 ℓ, ℓ [0, 1] 3 1 2 − ∈ 1 2 4 − ∈ the left hand side of the (Green’s formula) can be written as

t2 1 g1(iℓ)i dℓ + g1(ℓ + it2) dℓ+ t1 0 t2 1 g (1 + i(t + t ℓ))( i) dℓ + g (1 ℓ) dℓ. 1 1 2 − − − 1 − t1 0 Replacing g1(ζ +1) by g1(ζ)+1 and making a change of coordinate in the third integral, we obtain

1 1 i(t t )+ g (ℓ + it ) dℓ + g (1 ℓ) dℓ. − 2 − 1 1 2 − 1 − 0 0 Now we show that the above two integrals are in bounded distance of Lh(w2) and L (w ), as follows: − h 1 1 1 g1(ℓ + it2) dℓ Lh(w2) g1(ℓ + it2) Lh(w2) dℓ 0 − ≤ 0 | − | 1 = g (ℓ + it ) g (ℓ + it ) dℓ, | 1 2 − 1 1 2 | 0 1 5 sup g1′ (ζ) dℓ C5, ≤ 0 ζ [0,1]+it2 | | ≤ 4 ∈ for some ℓ [0, 1]. Similarly 1 ∈ 1 5 g1(1 ℓ) dℓ + Lh(w) C5. 0 − − ≤ 4 Now one infers parts (1) and (2) of the lemma by considering real part and imaginary part of Green’s formula).

4.1.6 Proof of main technical lemmas

Proof of Lemma 4.10. It is enough to prove the statement for small values of

α . That is because the sector f kn (S0) is contained in Dom f B(0, 4/27e4π). n n n n ⊂ 106 Therefore, it has uniformly bounded diameter. Now, one can choose a large

enough M to satisfy the inequality in the lemma once αn is not too small.

Let Ln denote the linearizing map Lfn corresponding to fn that can be obtained in Lemma 4.16. Consider the half-line

1/2αn γ(t) := F ⌊ ⌋(cp )+ it :[ 1/α 4C , ) C. n Fn − n − 7 ∞ →

By part (4) of Lemma 4.15,

1/2αn 1 1 Re Fn⌊ ⌋(cpFn ) [ 1/2αn C4(1 + log ), 1/2αn + C4(1 + log )]. ∈ ⌊ ⌋ − 2αn ⌊ ⌋ 2αn

Thus, for suﬃciently small αn, one can use Lemma 4.18, with r = 1/4, w1 =

1/2αn ⌊ ⌋ Fn (cpFn ) and w2 any point on γ, to conclude that 1 diam Re Ln(γ(t)) : t Dom γ 4C7, { − 2αn ∈ } ≤ Im L (γ( 1/α 4C )) 1/α . n − n − 7 ≤ − n

Hence, the set

B(Ln(γ(t)), 4C7 + 2) t Dom γ ∈ contains the half-strip

1 1 1 A := ζ C : 1/2 Re ζ +1/2, Im ζ . (4.9) { ∈ ⌊2αn ⌋ − ≤ ≤ ⌊2αn ⌋ ≥ −αn }

1 Now, by Lemma 4.17, image of this strip under Ln− must be contained in the set

B(γ(t),C5(4C7 + 2)), t Dom γ ∈ which is, by Lemma 4.15 part (4), a subset of the half-strip

1 1 B := ζ C : Re ζ C4(1 + log )+ C5(4C7 + 2), ∈ | − ⌊2αn ⌋| ≤ 2αn 1 1 Im ζ − 4C7 C4(1 + log ) C5(4C7 + 2) . ≥ αn − − 2αn − 107 0 By deﬁnition of Sn,

f kn (S0)= z :1/2 Re Φ (z) 3/2, Im Φ (z) 2 . n n { ∈ Pn ≤ n ≤ n ≥ − }

Equivariance relation, (Theorem 4.3–b) ), implies that

kn+ 1/2αn 1 0 fn ⌊ ⌋− (Sn) 1 1 = z n : 1/2 Re Φn(z) +1/2, Im Φn(z) 2 . { ∈ P ⌊2αn ⌋ − ≤ ≤ ⌊2αn ⌋ ≥ − }

1 1 Since Φ− = τ L− , to conclude the lemma, It is enough to bound diam τ (B). n n ◦ n n For small α , Lemma 4.14 with r = 1/4 applies and we obtain diam τ (B) n n ≤ π(2+4C7+C5(4C7+2)+3C4) Mαn, where M =4C1e . We have further shown that:

ζ A, τ (ζ) Mα (4.10) ∀ ∈ | n | ≤ n

Which will be used later.

Proof of Lemma 4.11. If α is large and w is also bounded below then one n | | can make choose the constant M large enough. So we only consider other cases.

First assume that αn is small so that the following argument works. Recall that η is an arbitrarily chosen inverse branch of Exp on . So we may n+1 Pn+1 assume that Re(η ( ) [0, kˆ] by Theorem 4.3. If we let ζ = η (w), n+1 Pn+1 ⊂ n+1 1 27 w 1 then Im ζ = − log | | . Now, let κ(n) := . 2π 4 ⌊ 2αn ⌋ It follows from Lemma 4.15 and 4.18 with r = 1/4 (for small enough αn)

1 that Ln− (ζ + κ(n)) satisﬁes the following:

1 1 3 Re Ln− (ζ + κ(n)) , 4αn ≤ ≤ 4αn

1 1 27 w 1 Im Ln− (ζ + κ(n)) − log | | 4C7 C4(1 + log ). ≥ 2π 4 − − 2αn

108 Now one uses Lemma 4.14 part (2), with r =1/4, to obtain

κ(n) 1 f (ψ (w)) = τ (L− (ζ + κ(n))) | n n+1 | | n n | −1 2παn Im Ln (ζ) 4C α e− ≤ 1 n 27C e(4C7+2)π α w αn . ≤ 1 n| | which proves the lemma in this case. The lemma for larger α and suﬃciently small w follows from compactness n | | of the class [α ]. Indeed, fn belongs to the class [α ] with αn [ε,α∗] for IS ∗ IS ∗ ∈ some ε and one can see that the associated map Fn converges geometrically to z z + 1 as Im z . This implies that the linearizing map L is → → ∞ n bounded away from a translation at points with large imaginary part. Now

1 one uses continuous dependence of linearizing map Lh− on the map Fh on a compact set [ε,α∗] [ 2, large number] to conclude that the translation × − constant must have a bounded absolute value. Therefore, if ζ = ηn+1(w), then

1 Im L− (ζ) Im ζ M ′ for some constant M ′. Like above argument this | n − | ≤ implies that for any choice of κ(n) [0, 1 k 1] we have ∈ αn − −

κ(n) αn f ψ (w) M ′′ α w , | n ◦ n+1 | ≤ n | | for some constant M ′′.

A corollary of our proof of Theorem 1.3 is the following.

Corollary 4.19. Let P (z)= z(1+ z)2, U be the domain deﬁned in (4.1), and

α be a non-Brjuno number in IrrN . If h is a rational map of the Riemann sphere with the following properties

h(0) = 0, h′(0) = 1, •

h(c) / U, if c is a critical point of h. • ∈ 109 Then the rational map g(z) := e2παi P h is non-linearizable at 0. ◦

Note that we do not assume in the above corollary that the corresponding Julia set has positive area.

Proof. Above conditions on h implies that g restricted to U belongs to [α ]. IS ∗ n All the sectors S0 are defined for g and satisfy the estimates in Proposition 4.12. Thus, the critical point which visits all the sectors, by definition, must accumulate on the fixed point. Hence, g is not linearizable at 0.

110 4.2 Measure and topology of the attractor

In this section we consider Lebesgue measure (area) and topology of the post- critical set of quadratic polynomials with a non-Brjuno multiplier of high return times.

n We will show that intersection of the sets Ω0 , which contains the post- critical set by Proposition 4.8, has area zero, by showing that it does not contain any Lebesgue density point. Strategy of our proof is to show that given any point z in this intersection, one can ﬁnd balls of arbitrarily small size but comparable to their distance to z in the complement of the intersection.

The balls will be introduced in domains of the renormalized maps fn and then transferred through our changes of coordinates to the dynamic plane of Pα. We will use the Koebe distortion theorem to derive required properties about shape, size, and the distance to z of the image balls.

4.2.1 Balls in the complement

The following lemma guarantees the complementary balls within the domain of each fn.

Lemma 4.20. There are positive constants δ and r∗ such that for every ζ C 1 ∈ with Im ζ 1 log 1 , and Exp(ζ) Ω0 for some integer n 1, there ≤ 2π αn+1 ∈ n+1 ≥ exists a line segment γ : [0, 1] C with γ (0) = ζ satisfying the following n → n properties:

0 0 (1) Exp B(γ (1),r∗) Ω = ∅, f Exp(B(γ (1),r∗)) Ω = ∅, n ∩ n+1 n+1 n ∩ n+1 (2) Exp B (B(γ (1),r∗) γ [0, 1]) Dom(f ) 0 , δ1 n ∪ n ⊆ n+1 \{ } (3) diam Re B (B(γ (1),r∗) γ [0, 1]) 1 δ , δ1 n ∪ n ≤ − 1 111 (4) mod B (B(γ (1),r∗) γ [0, 1]) (B(γ (1),r∗) γ ) δ . δ1 n ∪ n \ n ∪ n ≥ 1

1 1 Proof. First assume α min ′′ , ,ε , where ε is the constant n+1 ≤ { k +k 8(k+1) 0} 0 obtained in Lemma 4.15. Consider the line segment 1 ϑ(t) := t (2 + t/2)i : [2, ] C − 2αn+1 → between the two points 2 3i, and 1 (2+ 1 )i. For every t [2, 1 ], − 2αn+1 − 4αn+1 ∈ 2αn+1 under our assumption α 1 , we have n+1 ≤ 4(k+1) 1 B(ϑ(t), t/2) w C :0 Re(w) k, Im w 2 , ⊂{ ∈ ≤ ≤ α − ≤ − } n+1 (4.11) 1 B(ϑ(t), t/2)+1 w C :0 Re(w) k, Im w 2 . ⊂{ ∈ ≤ ≤ αn+1 − ≤ − } Similarly (when α 1/8k) one can see that n+1 ≤ 1 B(ϑ(t), 3t/4) w C :0 Re(w) k = Φn+1( n+1) ⊂{ ∈ ≤ ≤ αn+1 − } P which gives the following lower bound for conformal modulus: 1 3 mod (Φ ( ) B(ϑ(t), t/2)) log . (4.12) n+1 Pn+1 \ ≥ 2π 2

1 The idea of the proof is to show that lifts of Φn−+1(B(ϑ(t), t/2)) via Exp provide balls satisfying the required properties in the lemma. First we will

1 consider lifts of the curve Φ− ϑ, via Exp and show that they start from n+1 ◦ a bounded height and reach the needed height 1 log 1 . Then we consider 2π αn+1 1 lifts of Φn−+1(B(ϑ(t), t/2)) and show that they contain balls of a deﬁnite size. 1 1 Recall that Φ− = τ L− . By Lemma 4.17 we have n+1 n+1 ◦ n+1

1 1 L− (ϑ(2)) cv sup L− (2, 3) (1, 0) | n+1 − Fn+1 | ≤ | n+1|| − − | C √10. ≤ 5 Since τ maps the critical value cv to 4/27, one can see that every point n+1 Fn+1 − 1 1 in Exp− (Φn−+1(ϑ(2))) has imaginary part uniformly bounded above by some constant δ. 112 For the other end point, ϑ( 1 ) belongs to the half-strip A deﬁned in 2αn+1 (4.9). Thus by (4.10), we have

1 1 Φn−+1(ϑ( )) Mαn+1. | 2αn+1 | ≤

1 1 1 This implies that every point in Exp− (Φ− (ϑ( ))) has imaginary part n+1 2αn+1 bigger than 1 log 1 1 log 27M . 2π αn+1 − 2π 4 To transfer the balls, consider the map

1 η τ L− : Φ ( ) C (4.13) n+1 ◦ n+1 ◦ n+1 n+1 Pn+1 → where ηn+1 is an arbitrary inverse branch of Exp deﬁned on C minus a ray landing at 0.

We claim that there exists a constant M ′ such that derivative of the above map at every point t 2i C, t Dom ϑ, is at least M ′/t. By compactness of − ∈ ∈ the class [α ] and continuous dependence of linearizing map in the compact- IS ∗ open topology it is enough to prove the claim for values of t bigger than some constant (indeed when C (1+log t) t/2)). Also by Koebe distortion theorem, 4 ≤ it is enough to prove this for integer values of t bigger than that constant. For

1 t such t’s, Ln−+1(t)= Fn+1(cvFn+1 ). So by Lemma 4.15 part (4), we have

1 1 Im L− (t) C (1 + log t), and Re L− (t) t C (1 + log t). | n+1 | ≤ 4 | n+1 − | ≤ 4

Hence, by Lemma 4.17,

1 Im L− (t 2i) C (1 + log t)+2C , (4.14) | n+1 − | ≤ 4 5 1 Re L− (t 2i) t C (1 + log t)+2C . | n+1 − − | ≤ 4 5

Deﬁne the set Ot as follows:

O := ξ C : Im ζ C (1+log t)+2C , and Re ζ t C (1+log t)+2C t { ∈ | | ≤ 4 5 | − | ≤ 4 5} 113 The point t belongs to Ot, and by an explicit calculation one can see than

(η τ )′(t) 1/2t. n+1 ◦ n+1 ≥

As mod (C∗ O ) is bounded below independent of t (indeed, it is increasing \ t in terms of t), Koebe distortion theorem implies that there exists a constant

M ′′ such that for every ξ O , we have ∈ t

(η τ )′(ξ) M ′′/t. n+1 ◦ n+1 ≥

1 Since L− (t 2i) O , by (4.14), combining with Lemma 4.17 we have n+1 − ∈ t

1 M ′′ 1 (ηn+1 τn+1 Ln−+1)′(t 2i) . ◦ ◦ − ≥ C5 t

Again Koebe distortion theorem, using (4.12), implies that

1 ξ B(ϑ(t), t/2) we have, (η τ L− )′(ξ) M ′/t ∀ ∈ n+1 ◦ n+1 ◦ n+1 ≥

for some constant M ′ independent of t and αn+1. Therefore, image of the ball

B(ϑ(t), t/2) under the map (4.13) contains a ball of constant radius r∗ around

1 Exp pre-images of Φn−+1(ϑ(t)).

Domain of fn+1 contains the ball of radius .22, therefore, every point in C

with positive imaginary part is mapped into Dom fn+1 under Exp. To associate

a curve γn to the given point ζ, we consider the following two separate cases:

If Im ζ 1, by previous argument, there exists a point ζ′ in a lift of ≥ 1 Φ− (ϑ) (under Exp) which satisﬁes Re(ζ ζ′) 1/2 and Im(ζ ζ′) n+1 − ≤ − ≤ max δ, 1 log 27M . Deﬁne γ : [0, 1] C as the straight line segment be- { 2π 4 } n → tween ζ and ζ′ with γ (0) = ζ and γ (1) = ζ′. Thus, B(γ (1),r∗) γ [0, 1] n n n ∪ n projects into Dom fn+1. Moreover, if r∗ is chosen less than 1/4, we have

diam(Re(B(γ (1),r∗) γ [0, 1])) 3/4. n ∪ n ≤ 114 Hence, Exp is univalent on the 1/4 neighborhood of B(γ (1),r∗) γ [0, 1]. n ∪ n 1 Part (1) of the lemma follows from (4.11) and that (when α ′′ ) n+1 ≤ k +k

kn+1 1 − f j (S0 ) w : Im Φ (w) < 2 = ∅. n+1 n+1 ∩{ ∈ Pn+1 n+1 − } j=0 Parts (2) and (3) follows form deﬁnition.

Now assume that Im ζ is uniformly bounded above, or αn+1 is bounded below (which implies Im ζ is bounded above). In this case our argument is based on the compactness of the class [α ]. Indeed, there exists a δ′ > 0 IS ∗ such that

0 B ′ (Ω ) Dom f . (4.15) δ n+1 ⊂ n+1 Since ξ = Exp(ζ) is away from 0 and has uniformly bounded diameter, there exists a (uniformly bounded) real number s> 1, with

0 0 B(sξ,δ′/2) Ω = ∅, and f (B(sξ,δ′/2)) Ω = ∅. ∩ n+1 n+1 ∩ n+1

Now, deﬁne γ′(t) := tξ + (1 t)sξ : [0, 1] Dom f . The curve γ is − → n+1 n 1 deﬁned as the lift of γ′ starting at ζ. Thus, Exp− (B(sξ,δ′/2)) contains a ball of radius r∗ satisfying the lemma in this case as well.

Lemma 4.21. There exists a real constant δ δ such that for every ξ C 2 ≤ 1 ∈ with Exp(ξ) Ω0 , we have ∈ n+1 Exp(B(ξ, δ )) Dom(f ), • 2 ⊂ n+1

n Z, Exp(B(n, δ )) int f kn (S0) Ω0 . • ∀ ∈ 2 ⊂ n n ⊂ n

Proof. It follows from continuous dependence of the Fatou coordinate in the compact-open topology, that there exists a real constant δ > 0 such that for every n 0, ≥ B( 4/27, δ) f kn (S0)= ξ C : Im ξ > 2, 1/2 Re ξ 3/2 . − ⊂ n n { ∈ − ≤ ≤ } 115 The ﬁrst inclusion in the lemma follows from (4.15) and the second one follows from above observation.

For every integer n 1 and every integer j, with 0 j < 1 k, we deﬁne ≥ ≤ αn − curves In,j as follows

1 I := Φ− ξ C : Re ξ = j, Im ξ > 2 . n,j n { ∈ − }

0 0 Each In,j is a smooth curve contained in Ωn, and connects boundary of Ωn to 0. Also one can see that for every such n and j, every closed loop (image of a continuous curve with the same initial and terminal point) contained in

0 Ω I is contractible in C∗. This implies that there is a continuous inverse n \ n,j branch of Exp deﬁned on every Ω0 I . n \ n,j By compactness of the class [α ] there exists a positive integer k′ such IS ∗ that

1 j with 0 j < k, sup arg(z) 2πk′, (4.16) αn 0 ∀ ≤ − z Ω In,j ≤ ∈ n\ for every continuous branch of argument deﬁned on Ω0 I . We assume the n \ n,j following technical condition on αn’s

1 αn (4.17) ≤ 2k′ + k during this section.

4.2.2 Going down the renormalization tower

n Fix an arbitrary point z0 in n∞=0 Ω0 diﬀerent from 0. We associate a sequence of quadruples

(z ,w ,ζ , σ(i)) ∞ (4.18) { i i i }i=0

116 to z , where z and w are points in Dom (f ), ζ is a point in Φ ( ) and σ(i) is 0 i i i i i Pi a non-negative integer. This sequence will serve us as a guide to transfer the

balls in the previous lemma to the dynamic plane of f0. The sequence of quadruples (4.18) is deﬁned inductively as follows: Since

1/α0 k+kn 1 j z − − f (S0), we have one of the following two possibilities: 0 ∈ ∪j=0 0 0

Case I z , and one of the following two occurs: 0 ∈ P0

Re Φ (z ) [k′ +1/2, 1/α k], − 0 0 ∈ 0 −

Φ (z ) B(j, δ ) for some j =1, 2,...,k′ − 0 0 ∈ 2

Case II z , Re Φ (z ) [0,k′ +1/2), and Φ (z ) / B(j, δ ), for j = 0 ∈ P0 0 0 ∈ 0 0 ∈ 2 1, 2,...,k′. Or, z / . 0 ∈ P0

If case I occurs, deﬁne w0 := z0, σ(0) := 0, and ζ0 := Φ0(w0).

0 If case II occurs, let w S , and positive integer σ(0) k +k′ be such that 0 ∈ 0 ≤ 0 σ(0) f0 (w0)= z0. The point w0 satisfying this property is not necessarily unique, however, one can take any of them. The positive integer σ(0) is uniquely determined. Indeed when σ(0) k 1 or z is small enough, such w is ≤ 0 − | 0| 0 unique , otherwise, there are at most two choices for w0. The point ζ0 is

defined as Φ0(w0). This defines the first quadruple (z0,w0,ζ0, σ(0)).

1 Now, let z1 := Exp(ζ0). Since z0 belongs to Ω0, One can see that z1 belongs

1/α1 k+k1 1 j to − − f (S0). Thus, we can repeat the above process (replacing 0 by ∪j=0 1 1 1 in the above cases) to deﬁne the quadruple (z1,w1,ζ1, σ(1)) and so on. In

117 general, for every l 0, ≥

1/αl k+kl 1 j 0 zl = Exp(ζl 1), zl j=0 − − fl (Sl ), − ∈ ∪ σ(l) (4.19) fl (wl)= zl, Φl(wl) := ζl,

0 σ(l) k + k′ k′′ + k′ ≤ ≤ l ≤ where k′′ is a uniform bound on the integers kl. By deﬁnition of this sequence, for every n 0 we have ≥ 1 k′ +1/2 Re ζn k, or, ≤ ≤ αn − (4.20)

ζ B(j, δ ), for some j 1, 2,...,k′ . n ∈ 2 ∈{ }

The following lemma guarantees that some of ζj in the above sequence reach the balls provided in the previous lemma.

n Lemma 4.22. Assume that z ∞ Ω 0 , and α is a non-Brjuno num- 0 ∈ ∩n=0 0 \{ } ber in Irr . If ζ ∞ is the above sequence associated to z , then there are N { j}j=0 0 arbitrarily large positive integers m with

1 1 Im ζm log . ≤ 2π αm+1

To see this, we need the following lemma. Let D1 be a constant such that

D1 1 1 + C2 +4C7 + C4(1 + log ) αn+1 ≥ 4αn+1 αn+1 for every α [1/2, ), where the constants C ,C , and C were introduced n+1 ∈ ∞ 2 4 7 in Lemmas 4.15 and 4.18.

Lemma 4.23. There exists a positive constant D2 such that for every n> 0, we have

D1 if Im ζn+1 , ≥ αn+1 1 1 1 D2 then Im ζn+1 Im ζn log + . (4.21) ≤ αn+1 − 2παn+1 αn+1 αn+1

118 1 Proof. Given ζn+1 with Im ζn+1 D1/αn+1, there is an integer i with − ≥ αn+1 ≤ 1 1 1 1 i such that Re L− (ζn+1 + i) [ , +2]. By part (4) of ≤ αn+1 n+1 ∈ 2αn+1 2αn+1 Lemma 4.15, and Lemma 4.18, both with r =1/4, we obtain

1 1 Im Ln−+1(ζn+1 + i) Im(ζn+1 + i) 4C7 C4(1 + log ). ≥ − − αn+1

By our assumption on Im ζn+1, this implies that

1 1 Im Ln−+1(ζn+1 + i) + C2. ≥ 4αn+1 1 Now, one uses part (1) of Lemma 4.15, to conclude that i iterates of Ln−+1(ζn+1+ i) under Fn+1 stay in Θ(C2), and moreover,

1 i 1 Im Ln−+1(ζn+1) = Im Fn−+1(Ln−+1(ζn+1 + i))

1 i Im L− (ζ + i) ≥ n+1 n+1 − 4 1 1 Im ζn+1 4C7 C4(1 + log ) . ≥ − − αn+1 − 4αn+1 1 Using Lemma 4.14 with r =1/4 at Ln−+1(ζn+1) implies that

1 1 1 2παn+1 Im ζn+1 4C7 C4(1+log α ) 4α τ (L− (ζ )) 4C α e− − − n+1 − n+1 . | n+1 n+1 n+1 | ≤ 1 n+1 Hence, Φn+1(wn+1)= ζn+1 implies

1 w = Φ− (ζ ) | n+1| | n+1 n+1 | 1 1 2παn+1(+4C7+C4(1+log α )+ 4α ) 2παn+1 Im ζn+1 4C e n+1 n+1 α e− ≤ 1 n+1

2παn+1 Im ζn+1 Cα e− , ≤ n+1 for some constant C.

As wn+1 is mapped to zn+1 in a bounded number of iterates σ(n) under

f which belongs to a compact class, z C′ w for some constant n+1 | n+1| ≤ | n+1| C′. Therefore,

2π Im ζn 2πiζn 4/27e− = 4/27e− | − | 2παn+1 Im ζn+1 = z CC′α e− . | n+1| ≤ n+1 119 Multiplying by 27/4 and then taking log of both sides, one obtains Inequal-

ity (4.21) for some constant D2.

Proof of Lemma 4.22. Given integer ℓ 1, we will show that there exists ≥ m ℓ satisfying the inequality in the lemma. For arbitrary α, one of the ≥ following two occurs

( ) There exists a positive integer n ℓ such that for every j n , we have ∗ 0 ≥ ≥ 0 D1 Im ζj . ≥ αj

D1 ( ) There are inﬁnitely many integers j, j ℓ, with Im ζj < . ∗∗ ≥ αj

Assume conclusion of the lemma is not correct, that is, for every m greater than or equal to ℓ we have Im ζ > 1 log 1 . We will show that each of the m 2π αm+1 above cases leads to a contradiction. If ( ) holds, we can use Lemma 4.23 for every j n . So, for every integer ∗ ≥ 0 n bigger than n0, using Relation (4.21) repeatedly, we obtain

1 1 1 Im ζn Im ζn0 1 log ≤ αnαn 1 αn0 − − 2παnαn 1 αn0 αn0 − − 1 1 1 1 log log − 2παnαn 1 αn0+1 αn0+1 − 2παn αn − 1 1 1 + D2 + + + . (4.22) αnαn 1 αn0 αnαn 1 αn0 1 αn − − −

Let β 1 := 1, and βj := α0α1 αj, for every j 0. Using our contradiction − ≥ assumption and then multiplying both sides of the above inequality by 2πβn, we see

n 1 βj log 2πβn0 1 Im ζn0 1 +2πD2 (βn0 1 + βn0 + + βn 1) αj+1 ≤ − − − − j=n0 1 −

2π Im ζn0 1 +2πD2. ≤ − 120 Since n was an arbitrary integer, this contradicts α being a non-Brjuno number. Now assume ( ) holds. Let n < m n < m n < be an ∗∗ 1 2 ≤ 2 3 ≤ 3 increasing sequence of positive integers with the following properties

D1 For every integer j with mi j ni, we have Im ζj < • ≤ ≤ αj

D1 For every integer j with ni

Estimate (4.22) holds for j = n +1, n +2, , m 1, where Lemma 4.23 i i i+1 − can be used, and implies that for every i 2: ≥

mi+1 1 − 1 βj log 2πβni Im ζni +2πD2 βni + βni+1 + + βmi+1 2 . αj+1 ≤ − j=ni Hence,

∞ 1 1 1 βj log = βj log + βj log αj+1 αj+1 αj+1 j=m2 j; mi j

∞ ∞ 2πD1 βj 1 +2πD1 βni 1 +2πD2 βj − − j; mi j

4.2.3 Going up the renormalization tower

i ♯ i Recall the sectors − ( )− , for i = 1, 2,...,k , introduced in deﬁnition Cn ∪ Cn n 0 kn ♯ kn of the renormalization (for f ), where S = − ( )− . If k < k′ + 1, n n Cn ∪ Cn n 121 by our assumption (4.17) on k′, we can consider further pre-images for i = k +1, ,k′ +1 as n

i 1 kn − := Φ− (Φ ( − ) (i k )), Cn n n Cn − − n ♯ i 1 ♯ kn ( )− := Φ− (Φ (( )− ) (i k )). Cn n n Cn − − n

k′ 1 ♯ k′ 1 k′+1 Let denote the sector − − ( )− − , and observe that f : Dn Cn ∪ Cn n Dn → kn 0 fn (Sn). For every integer n 0, define the set ♮ as: ≥ Pn k′ ♮ := f j( ). Pn n Dn j=0 We define a map Φ♮ : ♮ C, using the dynamics of f , as follows. For n Pn → n ♮ j z , there is an integer j with 0 j k′ + 1, such that f (z) . Now, ∈ Pn ≤ ≤ n ∈ Pn let Φ♮ (z) := Φ (f j(z)) j. n n n − ♮ As Φn satisfies the Abel functional equation, one can see that Φn matches on the boundary of above sectors and gives a well defined holomorphic map on ♮ . The map Φ♮ is not univalent, however, it still satisfies the Abel Functional Pn n equation on ♮ . It has critical points at the critical point of f and its pre- Pn n ♮ ♮ images within . The k′ + 1 critical points of Φ are mapped to 0, 1,...,k′. Pn n ♮ The map Φn is a natural extension of Φn to a multi-valued holomorphic map on ♮ . However, the two maps Pn ∪ Pn ′ ′ k 1/αn+k k 1 − − Φ♮ : f j( ) C, and Φ : f j( ) C n n Dn → n n Dn → j=0 ′ j=k +1 provide a well-defined holomorphic map on every k′ + 1 consecutive sectors of the form f j( ). We denote this map by Φ♮ Φ . More precisely, for every l n Dn n ∐ n

122 with 0 l< 1 k, ≤ αn − k′ Φ♮ Φ : f l+j( ) C n ∐ n n Dn → j=0 is deﬁned as

♮ i Φ (z), if z f ( n) and i

Vn,Vn 1,...,V1 and holomorphic maps gn+1,gn,...,g1 satisfying the following − diagram

gn+1 gn gn−1 g1 0 n Vn Vn 1 V1 V0 := B(Ω0, 1) (4.23) A −−−→ −−−→ − −−−→ −−−→ where,

0 1 Vm = Ω Im,j(m), for some j(m) with 0 j(m) < k, m \ ≤ αm − 0 gn+1 : n Vn,g1 : V1 Ω0, A → → (4.24) gm : Vm Vm 1, for every m with 1 m n, → − ≤ ≤

gm(zm)= zm 1 for every m =1, 2, . . . , n. −

The idea is to use an inductive process to deﬁne pairs gi +1,Vi, starting with i = n and ending with i = 0.

Base step i = n

We have ζ and satisfies (4.20). As diam( ) 1 δ , and δ < δ , n ∈ An An ≤ − 1 2 1 there exists an integer j, with 0 j 1, such that ≤ ≤ 1 Re( n j) (0, k). A − ⊂ αn − 123 We define g : C as n+1 An → j+σ(n) 1 g (ζ) := f (Φ− (ζ j)). n+1 n n − By Lemma 4.20, Exp( j) is contained in Dom f . So Lemma 4.5 implies An − n+1 j+σ(n) 1 that fn is defined at every point in Φ− ( j), that is, the above map n An − is well defined. Using the same lemma, as j + σ(n) 1+ k + k′ by (4.19), we ≤ n conclude that g ( ) is contained in Ω0 . n An n Because j is contained in int Φ ( ), it does not intersect the vertical An − n Pn 1 line ξ C : Re ξ = 0 . Therefore, Φ− ( j) does not intersect the curve { ∈ } n An − j+σ(n) 1 I . One can see from this that, g ( ) = fn (Φ− ( j)) does not n,0 n+1 An n An − 1 intersect the curve In,j′ , where j′ is j + σ(n) module k. We define Vn ⌊ αn ⌋ − as Ω0 I ′ . n \ n,j Finally, by equivariance property of Φn,

j+σ(n) 1 σ(n) g (ζ )= f (Φ− (ζ j)) = f (w )= z . n+1 n n n n − n n n

Induction step

Assume (gi+1,Vi) is deﬁned and we want to deﬁne (gi,Vi 1). Since every closed −

loop in Vi is contractible in C∗, there exists an inverse branch of Exp, denoted

by ηi, deﬁned on Vi with ηi(zi)= ζi 1. Now consider the following two cases, − case i Re(η (V )) (1/2, ), i i ⊂ ∞ case ii Re(η (V )) ( , 1/2] = ∅. i i ∩ −∞

Case i

Since ζ η (V ) satisﬁes (4.20) and diam B (η (V )) k′ +1/2 by (4.16), n ∈ i i δ2 i i ≤ there exists an integer j, 0 j k′ + 1, with ≤ ≤ 1 1 1 Bδ2 (ηi(Vi)) j ξ C : Re ξ k . (4.25) − ⊂{ ∈ 4 ≤ ≤ αi−1 − − 2 } 124 We deﬁne g : B (η (V )) C as i δ2 i i → j+σ(i 1) 1 gi(ζ) := fi 1 − (Φi− 1(ζ j)), (4.26) − − − and let g (z) := g η(z). i i ◦

By Lemma 4.21, Exp(Bδ2 (ηi(Vi))) is contained in Dom fi. Thus, Lemma 4.5 j+σ(i 1) 1 and condition (4.17) implies that fi 1 − is deﬁned on Φi− 1(Bδ2 (ηi(Vi)) j), − − − that is, the above map is well deﬁned.

By equivariance property of Φi 1 (Theorem 4.3), we have −

j+σ(i 1) 1 σ(i 1) gi(zi)= fi 1 − (Φ−i 1(ηi(zi 1) j)) = fi 1− (wi 1)= zi 1. − − − − − − −

One also concludes from (4.25) that B (η (V )) j does not intersect the δ2 i i − vertical line ξ C : Re ξ =0 . Therefore, { ∈ }

j+σ(i 1) 1 gi(Bδ2 (η(Vi)) j)= fi 1 − (Φi− 1(Bδ2 (ηi(Vi)) j)) − − − −

1 does not intersect the curve Ii 1,j′ , where j′ is j + σ(i 1) module k. − − ⌊ αi−1 ⌋ − 0 ′ Hence, by deﬁning Vi 1 := Ωi 1 Ii 1,j , we have − − \ −

gi(Bδ2 (ηi(Vi)) j) Vi 1, (4.27) − ⊂ − which will be used later.

Case ii

Because diam(ηi(Vi)) k′ and ηi(Vi) contains ζi 1 which satisﬁes (4.20), we ≤ − must have ζi 1 B(j, δ2) for some j in 1, 2,...,k′ . − ∈ { } We claim that

B (η (V )) 0, 1, 2,..., k′ = ∅. (4.28) δ2 i i ∩{ − − − } 125 As Exp(Z)= 4/27, it is equivalent to see that η ( 4/27) / 0, 1,..., k′ . − i − ∈{ − − } But by inclusion in the Lemma (4.21), for every integer n, we have

Exp(B(n, δ )) int V . 2 ⊂ i

This implies that η ( 4/27) 1, 2, ,k′ . i − ∈{ }

The set Bδ2 (ηi(Vi)) has diameter strictly less than k′ + 1, so it can intersect at most k′ +1 vertical strips of width 1. More precisely, Bδ2 (ηi(Vi)) is contained in the k′ + 1 consecutive sets in the list

♮ ♮ ♮ k′ 1 Φi 1( i 1), Φi 1(fi 1( i 1)),..., Φi 1(fi −1 ( i 1)), − D − − − D − − − D − k′ 2k′+1 Φi 1(fi 1( i 1)),..., Φi 1(fi 1 ( i 1)). − − D − − − D −

♮ Thus, by the above argument about Φi 1 Φi 1, and that every closed loop in − ∐ − ♮ Bδ2 (ηi(Vi)) is contractible in the complement of the critical values of Φi 1 Φi 1, − ∐ − there exists an inverse branch of this map, denoted by gi, deﬁned on Bδ2 (ηi(Vi)). We let 0 gi(z) := gi(ηi(z)) : Vi Ωi 1. → −

In this case σ(i 1)=0, Φi 1(wi 1)= ζi 1, and wi 1 = zi 1. So gi(zi)= zi 1. − − − − − − −

Like previous case, one can see that gi(Bδ2 (ηi(Vi))) does not intersect the

0 curve Ii 1,j for j = sup Re(Bδ2 (ηi(Vi))) + 1. We can deﬁne Vi 1 := Ωi 1 and − { } − − obtain gi : Vi Vi 1. Indeed, we have → −

gi(Bδ2 (ηi(Vi))) Vi 1. (4.29) ⊂ −

This ﬁnishes deﬁnition of the domains and maps satisfying (4.24).

Each domain Vn,Vn 1 ...,V0, is a hyperbolic Riemann surface. Let ρi de- − note the Poincar´emetric on V , that is, ρ (z) dz is the complete metric of i i | | 126 constant negative curvature on Vi. Similarly, ρn+1 denotes the Poincare metric on . The following two lemmas are natural consequence of our construction An of the chain (4.23).

Lemma 4.24. Each map gi :(Vi, ρi) (Vi 1, ρi 1), for i = n, n 1,..., 1, is → − − − uniformly contracting. More precisely, for every z V , we have ∈ i

ρi 1(gi(z)) gi′(z) δ3 ρi(z), − | | ≤

2k′+1 for δ = ′ . 3 2k +1+δ2

Proof. Letρ ˜ (z) dz andρ ˆ (z) dz denote the Poincar´emetric on the domains i | | i | |

ηi(Vi) and Bδ2 (ηi(Vi)), respectively. By deﬁnition of gi and properties (4.27) and (4.29) we can decompose the map gi :(Vi, ρi) (Vi 1, ρi 1) as follows: → − −

ηi inc. g˜i (Vi, ρi) / (ηi(Vi), ρ˜i) / (Bδ2 (ηi(Vi)), ρˆi) / (Vi 1, ρi 1). − −

By Schwartz-Pick lemma the ﬁrst map, and the last map are non-expanding, i.e.,

ρ˜i(ηi(ζ)) ηi′(ζ) ρi(ζ), and ρi 1(˜gi(ζ)) g˜i′(ζ) ρˆi(ζ). | | ≤ − | | ≤ To show that the inclusion map is uniformly contracting in the respective

metrics, ﬁx an arbitrary point ζ0 in ηi(Vi), and deﬁne δ H(ζ) := ζ +(ζ ζ ) 2 : η (V ) C. − 0 (ζ ζ +2k + 1) i i → − 0 ′ Since diam η (V ) k′, we have Re(ζ ζ )) k′ for every ζ η (V ) and also i i ≤ | − 0 | ≤ ∈ i i ζ ζ0 H(ζ )= ζ . This implies that − ′ < 1. Thus, 0 0 ζ ζ0+2k +1 | − | ζ ζ H(ζ) ζ = δ − 0 < δ , | − | 2|ζ ζ +2k +1| 2 − 0 ′ which implies that H(ζ) is a holomorphic map from ηi(Vi) into Bδ2 (ηi(Vi)). By

Schwartz-Pick lemma, H is non-expanding. In particular at ζ0, we obtain

δ2 ρ˜i(ζ0) H′(ζ0) =ρ ˜i(ζ0)(1 + ) ρˆi(ζ0). | | 2k′ +1 ≤ 127 2k′+1 That is,ρ ˆi(ζ0) δ3 ρ˜i(ζ0) for δ3 = ′ < 1. Putting all this together gives ≤ 2k +1+δ2 the inequality in the lemma.

Lemma 4.25. There exists a positive constant δ4 such that for every i = 1, 2,...,n +1, the following holds

The map gi : Vi Vi 1 is univalent or has only one simple critical point • → − .

The map gi : Vi Vi 1 is univalent on the hyperbolic ball • → −

B (z , δ ) := z V d (z, z ) < δ . ρi i 4 { ∈ i | ρi i 4}

Proof. Each map gi is composition of at most four maps; ηi, a translation by

1 j+σ(i 1) an integer j, Φi− 1, and fi 1 − . The ﬁrst three maps are univalent. The map − − j+σ(i 1) 1 fi 1 − is univalent or has at most one simple critical in Φi− 1(ηi(Vi) j). To − − − j+σ(i 1) see this, ﬁrst note that the critical points of fi 1 − are −

1 j σ(i 1) − − − cpfi−1 , fi− 1(cpfi−1 ), fi 1 (cpfi−1 ) . { − − }

All of them are non-degenerate and, by our technical assumption, j +σ(i 1) − ≤ 1 2k′ + 1. If Φi− 1(ηi(Vi) j) contains more than one point in the above list, by − − equivariance property of Φi 1, there must be a pair of points ξ, ξ +m (for some − integer m) in η (V ) j. As this set is a lift of a simply connected domain in i i − C∗ under Exp, that is not possible.

The maps gi introduced in case ii are univalent, therefore, to see the second part, we only need to consider maps introduced in case i in the above inductive process. First we claim that there exists a real constant δ > 0, such that the ball

z Vi 1 : dρi−1 (z, zi 1) < δ { ∈ − − }

128 is simply connected and does not contain critical value of gi (if there is any critical value).

Assuming the claim for a moment, one can take δ4 := δ. Because, by

the previous lemma, image of Bρi (zi, δ4) is contained in the above ball. As

Bρi−1 (zi 1, δ) is simply connected and does not contain any critical value, one − can ﬁnd a univalent inverse branch for gi on this ball. Therefore, gi is univalent on the ball Bρi (zi, δ4).

To prove the claim, note that by deﬁnition (4.26) of gi 1, and condition −

0 j k′ + 1, a possible critical value of gi 1 can only be one of ≤ ≤ −

k′ 4/27, fi 1( 4/27),...,fi 1( 4/27). − − − − −

1 First we show that if cvgi−1 belongs to Φi− 1(B(l, δ2)) for some l 1, 2,...,k′ , − ∈{ } then zi 1 does not belong to this set. To see this, we consider case I and case − II in the deﬁnition of quadruples (4.18) separately.

If case I holds, then we have σ(i 1) = 0, zi 1 = wi 1, and ζi 1 = − − − − Φi 1(zi 1). If Re ζi 1 k′ +1/2, then zi 1 does not belong to any of the balls − − − ≥ − 1 Φi− 1(B(l, δ2)) for l 1, 2,...,k′ . If ζi 1 B(l, δ2) for some l 1, 2 ...,k′ , − ∈{ } − ∈ ∈{ } 1 then by (4.26) there is no critical value of gi 1 in any of Φi− 1(B(l, δ2)). − −

If case II holds, then, by deﬁnition of the quadruples, zi 1 does not belong −

to any of Φi 1(B(l, δ2)) for l =1, 2,...,k′. − 1 Finally, we need to show that each set Φi− 1(B(l, δ2)), for l = 1, 2 ...,k′, − contains a hyperbolic ball of radius δ independent of l and i. Fix such an l, and observe that

mod Φi 1( i 1) B(l, δ2) l it t [0, 2] c − P − \{ ∪{ − | ∈ } ≥ 1 for some constant c > 0. By Koebe distortion theorem for Φi− 1, we conclude −

129 that 1 Euclidean diameter (Φi− 1(B(l, δ2))) − 1 c′ Euclidean distance between Φi− 1(l) and ∂Vi 1 ≥ − − As ρi 1 is proportional to one over distance to the boundary in Vi 1, the − − 1 set Φi− 1(B(l, δ2)) contains a round hyperbolic ball of radius uniformly bounded − below. It is clear that each of these balls is simply connected.

Let denote the map Gn := g g g : Ω0. Gn 1 ◦ 2 ◦ n+1 An → 0

Recall that γn is the line segment obtained in Lemma 4.20, and γn(0) = ζn. So, (γ (0)) = z . The following lemma guarantees that safely transfers Gn n 0 Gn the ball from level n + 1 to the dynamic plane of f0.

Lemma 4.26. There exists a positive constant D such that for every in- 3 Gn troduced above, there is a positive constant rn with the following properties,

n+1 (1) (B(γ (1),r∗)) Ω = ∅, Gn n ∩ 0

(2) B( (γ (1)),r ) (B(γ (1),r∗)), and (γ (1)) z D r , Gn n n ⊂ Gn n |Gn n − 0| ≤ 3 n (3) r D (δ )n. n ≤ 3 3 Proof.

Part (1): By Lemma 4.20, for every z B(γ (1),r∗) we have ∈ n Exp(z) / Ω0 , and f (Exp(z)) / Ω0 . ∈ n+1 n+1 ∈ n+1 We claim that this implies

g (z) / Ω1 , and f (g (z)) / Ω1 , n+1 ∈ n n n+1 ∈ n where 1/αn (kn+1+1/αn+1 k 1)+1 ⌊ ⌋ − − 1 i 0 Ωn = fn(ψn+1(Sn+1)). i=0 130 That is because if g (z) Ω1 , then by deﬁnition of renormalization and n+1 ∈ n deﬁnition of Ω1 , there is a Ω1 , and b Ω1 , such that f i1 (a) = n ∈ Pn ∩ n ∈ Pn ∩ n n i2 gn+1(z), fn (gn+1(z)) = b, Exp(Φn(a)) = z, and Exp(Φn(b)) = fn+1(z) for non- negative integers i and i . One can see from this that Exp(Φ (a)) = z Ω0 1 2 n ∈ n+1 and Exp(Φ (b)= f (z) Ω0 which contradicts our assumption. n n+1 ∈ n+1 The same argument implies the following statement for every i = n, n − 1,..., 1,

n i+1 n i+1 If w / Ω − , and f (w) / Ω − ∈ i i ∈ i n i+2 n i+2 then gi(w) / Ωi −1 , and fi 1(gi(w)) / Ωi −1 ∈ − − ∈ −

k where Ωl is deﬁned accordingly. One infers from these, with an induction argument, that (z) / Ω0 . Gn+1 ∈ n+1

Part (2): It follows from part (4) of Lemma 4.20 that B (γ (1),r∗) γ [0, 1] n ∪ n has hyperbolic diameter (with respect to ρ in ) uniformly bounded by n+1 An some constant C (independent of n). Let l be the smallest non-negative integer with C (δ )l δ /2. 3 ≤ 4 We decompose the map into two maps Gn+1

1 2 n+1 := gn l+1 gn l+2 gn+1 and n+1 := g1 gn l+2 gn l. G − ◦ − ◦◦ G ◦ − ◦◦ − By Lemma 4.24 and our choice of l, we have

1 n+1(B(γn(1),r∗) γn[0, 1]) Bρn−l (zn l, δ4/2). G ∪ ⊆ −

Since each map gi is uniformly contracting and univalent on Bρi (zi, δ4), by

2 Lemmas 4.24 and 4.25, n+1 is univalent on Bρn−l (zn l, δ4). Moreover, by G − 1 Koebe distortion theorem, it has bounded distortion on (B(γ (1),r∗) Gn+1 n ∪ γn[0, 1]). 131 We claim that 1 belongs to a compact class. That is because it is Gn+1 composition of l maps (uniformly bounded independent of n) gi, for i = n + 1, , n l + 1, where each of them is composition of three maps as − g = f σ(i)+j g˜ (η j). i i ◦ i ◦ i −

The map ηi is univalent on Vi and, by Koebe distortion theorem, has uniformly

bounded distortion on sets of bounded hyperbolic diameter. The mapg ˜i ex-

tends over the larger set Bδ(ηi(Vi)) (see Equations (4.27) and (4.29)), so it also σ(i)+j has uniformly bounded distortion. Finally, fi is a ﬁnite iterate of a map in a compact class. Putting all these together, one infers that Euclidean diameter of the domain

1 (B(γ (1),r∗)) is proportional to the Euclidean distance between two points Gn+1 n 1 (γ (1)) and z = (γ (0)). Similarly, 1 (B ∗ ) contains a round ball Gn+1 n 0 Gn+1 n Gn+1 r of Euclidean radius proportional to its diameter. By previous argument about 2 , this ﬁnishes part (2) of the lemma. Gn+1

Part (3): First observe that (B(γ (1),r∗)) is contained in the compact Gn+1 n 0 subset Ω0 of V0 where the Euclidean and the hyperbolic metrics are proportional. The uniform contraction in Lemma 4.24 with respect to the hyperbolic metric implies the statement in this part.

4.2.4 Proof of main results

Corollary 4.27. There exists a positive integer N such that for every non- Brjuno number α Irr the post-critical set of P has area zero. ∈ N α Proof. If (P )= (f ) has positive area, by Lebesgue density point theo- PC α PC 0 rem, for almost every point z in this set we must have area(B(z,r) (f )) lim ∩ PC 0 =1. r 0 → areaB(z,r) 132 Let z be an arbitrary point diﬀerent from zero in (f ). By Proposi- 0 PC 0 tion 4.8, z is contained in the intersection of Ωn, for n = 0, 1, . Thus, we 0 0 can deﬁne the sequence of Quadruples (4.18). Lemma 4.22 provides us with

an strictly increasing sequence of integers ni for which we have 1 1 Im ζni log . ≤ 2π αni+1

With Lemma 4.20 at levels ni, we obtain curves γni and balls B(γni (1),r∗) enjoying the properties in that lemma. One introduces the sequence Gni+1

which, by Lemma 4.26, provides us with a sequence of balls B(γni (1),rni ) satisfying

B(γ (1),r ) Ω0 = ∅, (γ (1)) z D r , and r 0. ni ni ∩ ni+1 |Gni+1 ni − 0| ≤ 3 ni ni → With s := r + D r we have i ni 3 ni 2 2 area (B(z0,si)) (f0)) π(si) π(rni ) ∩ PC − 2 area (B(z0,si)) ≤ π(si) 2 (D3) +2D3 2 ≤ (D3) +2D3 +1 < 1.

which contradicts z being a Lebesgue density point of (f ). 0 PC 0 Corollary 4.28. There exists a positive integer N such that, if α Irr is a ∈ N non-Brjuno number, then Lebesgue almost every point in the Julia set of Pα is non-recurrent. In particular, there is no ﬁnite absolutely continuous (with respect to the Lebesgue measure) invariant measure supported on the Julia set.

Proof. By Propositions 4.8 and 4.2 almost every point in the complement of

n n Ω0 is non- recurrent. As area Ω0 shrinks to zero, we conclude the first part in the lemma. The second part follows from the first part and the Poincaré recurrence theorem. 133 Corollary 4.29. There exists a positive integer N such that for every non- Brjuno number α Irr the post-critical set of P is connected. ∈ N α

0 0 Proof. We claim that (P ) = ∞ Ω . As each set Ω is connected and PC α n=0 n n intersection of a nest of connected sets is connected, the corollary follows from this claim. To prove the claim, let z = 0 be an arbitrary point in the above intersection. We can build the sequence of Quadruples (4.18) corresponding to z.

By Lemma 4.22, there is an inﬁnite sequence of positive integers ni satisfying Im ζ 1 log 1 . It follows from proof of Lemma 4.20 that there exists ni 2π α +1 ≤ ni 1 1 a point ζn′ i in the lift ηni (Φ−ni ( 2α )) such that, Re(ζni ζn′ i ) 1/2, and ⌊ ni ⌋ | − | ≤ ζ ζ′ is uniformly bounded. One transfers these two point to the dynamic | ni − ni | plane of f by and concludes from Lemma 4.24 that (ζ′ ) z 0 Gni+1 |Gni+1 ni − | 1 1 goes to zero as n tends to inﬁnity. The point Φ− ( ) ) belongs to the i ni+1 2α +1 ⌊ ni ⌋

orbit of critical value of fni+1, therefore by deﬁnition of renormalization, see

Lemma 4.5, (ζ′ ) belongs to the orbit of the critical value of f . Thus, Gni+1 ni+1 0 (ζ′ ) (f ) and converges to z. This ﬁnishes proof of the claim. Gni+1 ni+1 ∈ PC 0 A corollary of above proof is the following:

Corollary 4.30. There are positive constants M, N, and < 1 such that for every α Irr and every z Ωn+1 we have ∈ N ∈ 0

P qn (z) z Mn. α − ≤

In particular this holds on the post-critical set.

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